Properties

Label 40-177e20-1.1-c2e20-0-0
Degree $40$
Conductor $9.109\times 10^{44}$
Sign $1$
Analytic cond. $4.63624\times 10^{13}$
Root an. cond. $2.19611$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 20·4-s − 8·7-s + 30·9-s + 158·16-s + 16·17-s − 60·19-s − 200·25-s − 160·28-s − 60·29-s + 600·36-s + 28·41-s − 316·49-s − 8·53-s − 152·59-s − 240·63-s + 522·64-s + 320·68-s + 92·71-s − 1.20e3·76-s − 420·79-s + 495·81-s − 4.00e3·100-s − 620·107-s − 1.26e3·112-s − 1.20e3·116-s − 128·119-s + 502·121-s + ⋯
L(s)  = 1  + 5·4-s − 8/7·7-s + 10/3·9-s + 79/8·16-s + 0.941·17-s − 3.15·19-s − 8·25-s − 5.71·28-s − 2.06·29-s + 50/3·36-s + 0.682·41-s − 6.44·49-s − 0.150·53-s − 2.57·59-s − 3.80·63-s + 8.15·64-s + 4.70·68-s + 1.29·71-s − 15.7·76-s − 5.31·79-s + 55/9·81-s − 40·100-s − 5.79·107-s − 11.2·112-s − 10.3·116-s − 1.07·119-s + 4.14·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 59^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 59^{20}\right)^{s/2} \, \Gamma_{\C}(s+1)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(3^{20} \cdot 59^{20}\)
Sign: $1$
Analytic conductor: \(4.63624\times 10^{13}\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{177} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 3^{20} \cdot 59^{20} ,\ ( \ : [1]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3449384622\)
\(L(\frac12)\) \(\approx\) \(0.3449384622\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - p T^{2} )^{10} \)
59 \( 1 + 152 T + 18302 T^{2} + 1459304 T^{3} + 109357965 T^{4} + 7084877840 T^{5} + 8624620440 p T^{6} + 9440238320 p^{2} T^{7} + 10696468086 p^{3} T^{8} + 10413087680 p^{4} T^{9} + 10884724444 p^{5} T^{10} + 10413087680 p^{6} T^{11} + 10696468086 p^{7} T^{12} + 9440238320 p^{8} T^{13} + 8624620440 p^{9} T^{14} + 7084877840 p^{10} T^{15} + 109357965 p^{12} T^{16} + 1459304 p^{14} T^{17} + 18302 p^{16} T^{18} + 152 p^{18} T^{19} + p^{20} T^{20} \)
good2 \( 1 - 5 p^{2} T^{2} + 121 p T^{4} - 1101 p T^{6} + 16697 T^{8} - 27359 p^{2} T^{10} + 638337 T^{12} - 1686389 p T^{14} + 8176355 p T^{16} - 18292811 p^{2} T^{18} + 303708557 T^{20} - 18292811 p^{6} T^{22} + 8176355 p^{9} T^{24} - 1686389 p^{13} T^{26} + 638337 p^{16} T^{28} - 27359 p^{22} T^{30} + 16697 p^{24} T^{32} - 1101 p^{29} T^{34} + 121 p^{33} T^{36} - 5 p^{38} T^{38} + p^{40} T^{40} \)
5 \( ( 1 + 4 p^{2} T^{2} + 54 T^{3} + 4784 T^{4} + 2048 T^{5} + 153714 T^{6} - 24872 T^{7} + 3760957 T^{8} - 2452838 T^{9} + 86828968 T^{10} - 2452838 p^{2} T^{11} + 3760957 p^{4} T^{12} - 24872 p^{6} T^{13} + 153714 p^{8} T^{14} + 2048 p^{10} T^{15} + 4784 p^{12} T^{16} + 54 p^{14} T^{17} + 4 p^{18} T^{18} + p^{20} T^{20} )^{2} \)
7 \( ( 1 + 4 T + 26 p T^{2} + 706 T^{3} + 20301 T^{4} + 64858 T^{5} + 1627341 T^{6} + 4395154 T^{7} + 103092678 T^{8} + 242700046 T^{9} + 5496972338 T^{10} + 242700046 p^{2} T^{11} + 103092678 p^{4} T^{12} + 4395154 p^{6} T^{13} + 1627341 p^{8} T^{14} + 64858 p^{10} T^{15} + 20301 p^{12} T^{16} + 706 p^{14} T^{17} + 26 p^{17} T^{18} + 4 p^{18} T^{19} + p^{20} T^{20} )^{2} \)
11 \( 1 - 502 T^{2} + 182661 T^{4} - 49609034 T^{6} + 1019503908 p T^{8} - 2179648042490 T^{10} + 375092432998995 T^{12} - 58362606555507626 T^{14} + 8338624859801251671 T^{16} - \)\(11\!\cdots\!12\)\( T^{18} + \)\(13\!\cdots\!80\)\( T^{20} - \)\(11\!\cdots\!12\)\( p^{4} T^{22} + 8338624859801251671 p^{8} T^{24} - 58362606555507626 p^{12} T^{26} + 375092432998995 p^{16} T^{28} - 2179648042490 p^{20} T^{30} + 1019503908 p^{25} T^{32} - 49609034 p^{28} T^{34} + 182661 p^{32} T^{36} - 502 p^{36} T^{38} + p^{40} T^{40} \)
13 \( 1 - 1350 T^{2} + 890761 T^{4} - 381297438 T^{6} + 705860804 p^{2} T^{8} - 29555113366886 T^{10} + 6287781440837163 T^{12} - 1253627755704023602 T^{14} + \)\(24\!\cdots\!23\)\( T^{16} - \)\(47\!\cdots\!72\)\( T^{18} + \)\(49\!\cdots\!88\)\( p^{2} T^{20} - \)\(47\!\cdots\!72\)\( p^{4} T^{22} + \)\(24\!\cdots\!23\)\( p^{8} T^{24} - 1253627755704023602 p^{12} T^{26} + 6287781440837163 p^{16} T^{28} - 29555113366886 p^{20} T^{30} + 705860804 p^{26} T^{32} - 381297438 p^{28} T^{34} + 890761 p^{32} T^{36} - 1350 p^{36} T^{38} + p^{40} T^{40} \)
17 \( ( 1 - 8 T + 1816 T^{2} - 6228 T^{3} + 1466195 T^{4} + 2814502 T^{5} + 701432321 T^{6} + 5681420338 T^{7} + 236215639120 T^{8} + 3200569946500 T^{9} + 68840644845798 T^{10} + 3200569946500 p^{2} T^{11} + 236215639120 p^{4} T^{12} + 5681420338 p^{6} T^{13} + 701432321 p^{8} T^{14} + 2814502 p^{10} T^{15} + 1466195 p^{12} T^{16} - 6228 p^{14} T^{17} + 1816 p^{16} T^{18} - 8 p^{18} T^{19} + p^{20} T^{20} )^{2} \)
19 \( ( 1 + 30 T + 2097 T^{2} + 58432 T^{3} + 2236518 T^{4} + 56806098 T^{5} + 1552758742 T^{6} + 36233869870 T^{7} + 41760314703 p T^{8} + 16961494042914 T^{9} + 318898249393354 T^{10} + 16961494042914 p^{2} T^{11} + 41760314703 p^{5} T^{12} + 36233869870 p^{6} T^{13} + 1552758742 p^{8} T^{14} + 56806098 p^{10} T^{15} + 2236518 p^{12} T^{16} + 58432 p^{14} T^{17} + 2097 p^{16} T^{18} + 30 p^{18} T^{19} + p^{20} T^{20} )^{2} \)
23 \( 1 - 5958 T^{2} + 18026833 T^{4} - 36663484032 T^{6} + 56127223009238 T^{8} - 68701771540253066 T^{10} + 69739236701940349758 T^{12} - \)\(60\!\cdots\!86\)\( T^{14} + \)\(44\!\cdots\!21\)\( T^{16} - \)\(28\!\cdots\!50\)\( T^{18} + \)\(16\!\cdots\!98\)\( T^{20} - \)\(28\!\cdots\!50\)\( p^{4} T^{22} + \)\(44\!\cdots\!21\)\( p^{8} T^{24} - \)\(60\!\cdots\!86\)\( p^{12} T^{26} + 69739236701940349758 p^{16} T^{28} - 68701771540253066 p^{20} T^{30} + 56127223009238 p^{24} T^{32} - 36663484032 p^{28} T^{34} + 18026833 p^{32} T^{36} - 5958 p^{36} T^{38} + p^{40} T^{40} \)
29 \( ( 1 + 30 T + 151 p T^{2} + 170670 T^{3} + 10389068 T^{4} + 423361222 T^{5} + 17553493628 T^{6} + 653450347678 T^{7} + 22299617133673 T^{8} + 722949440473712 T^{9} + 21511792590651646 T^{10} + 722949440473712 p^{2} T^{11} + 22299617133673 p^{4} T^{12} + 653450347678 p^{6} T^{13} + 17553493628 p^{8} T^{14} + 423361222 p^{10} T^{15} + 10389068 p^{12} T^{16} + 170670 p^{14} T^{17} + 151 p^{17} T^{18} + 30 p^{18} T^{19} + p^{20} T^{20} )^{2} \)
31 \( 1 - 11322 T^{2} + 61135393 T^{4} - 209308330836 T^{6} + 508918901744054 T^{8} - 932077214269053926 T^{10} + \)\(13\!\cdots\!18\)\( T^{12} - \)\(15\!\cdots\!90\)\( T^{14} + \)\(14\!\cdots\!65\)\( T^{16} - \)\(12\!\cdots\!70\)\( T^{18} + \)\(11\!\cdots\!42\)\( T^{20} - \)\(12\!\cdots\!70\)\( p^{4} T^{22} + \)\(14\!\cdots\!65\)\( p^{8} T^{24} - \)\(15\!\cdots\!90\)\( p^{12} T^{26} + \)\(13\!\cdots\!18\)\( p^{16} T^{28} - 932077214269053926 p^{20} T^{30} + 508918901744054 p^{24} T^{32} - 209308330836 p^{28} T^{34} + 61135393 p^{32} T^{36} - 11322 p^{36} T^{38} + p^{40} T^{40} \)
37 \( 1 - 15444 T^{2} + 121258654 T^{4} - 642109916454 T^{6} + 2565538855845221 T^{8} - 8203449137311550966 T^{10} + \)\(21\!\cdots\!21\)\( T^{12} - \)\(48\!\cdots\!42\)\( T^{14} + \)\(94\!\cdots\!54\)\( T^{16} - \)\(15\!\cdots\!58\)\( T^{18} + \)\(23\!\cdots\!14\)\( T^{20} - \)\(15\!\cdots\!58\)\( p^{4} T^{22} + \)\(94\!\cdots\!54\)\( p^{8} T^{24} - \)\(48\!\cdots\!42\)\( p^{12} T^{26} + \)\(21\!\cdots\!21\)\( p^{16} T^{28} - 8203449137311550966 p^{20} T^{30} + 2565538855845221 p^{24} T^{32} - 642109916454 p^{28} T^{34} + 121258654 p^{32} T^{36} - 15444 p^{36} T^{38} + p^{40} T^{40} \)
41 \( ( 1 - 14 T + 9824 T^{2} - 242374 T^{3} + 46717743 T^{4} - 1624216260 T^{5} + 146172069565 T^{6} - 6124161123484 T^{7} + 343318573182348 T^{8} - 14982774539742700 T^{9} + 641824765638483150 T^{10} - 14982774539742700 p^{2} T^{11} + 343318573182348 p^{4} T^{12} - 6124161123484 p^{6} T^{13} + 146172069565 p^{8} T^{14} - 1624216260 p^{10} T^{15} + 46717743 p^{12} T^{16} - 242374 p^{14} T^{17} + 9824 p^{16} T^{18} - 14 p^{18} T^{19} + p^{20} T^{20} )^{2} \)
43 \( 1 - 16474 T^{2} + 141759861 T^{4} - 834293490206 T^{6} + 3736863584815476 T^{8} - 13498651110051376286 T^{10} + \)\(95\!\cdots\!05\)\( p T^{12} - \)\(10\!\cdots\!78\)\( T^{14} + \)\(24\!\cdots\!71\)\( T^{16} - \)\(51\!\cdots\!88\)\( T^{18} + \)\(99\!\cdots\!72\)\( T^{20} - \)\(51\!\cdots\!88\)\( p^{4} T^{22} + \)\(24\!\cdots\!71\)\( p^{8} T^{24} - \)\(10\!\cdots\!78\)\( p^{12} T^{26} + \)\(95\!\cdots\!05\)\( p^{17} T^{28} - 13498651110051376286 p^{20} T^{30} + 3736863584815476 p^{24} T^{32} - 834293490206 p^{28} T^{34} + 141759861 p^{32} T^{36} - 16474 p^{36} T^{38} + p^{40} T^{40} \)
47 \( 1 - 17518 T^{2} + 156083133 T^{4} - 960261674600 T^{6} + 4655644696317798 T^{8} - 19025559809330919362 T^{10} + \)\(67\!\cdots\!54\)\( T^{12} - \)\(45\!\cdots\!78\)\( p T^{14} + \)\(60\!\cdots\!49\)\( T^{16} - \)\(15\!\cdots\!06\)\( T^{18} + \)\(35\!\cdots\!02\)\( T^{20} - \)\(15\!\cdots\!06\)\( p^{4} T^{22} + \)\(60\!\cdots\!49\)\( p^{8} T^{24} - \)\(45\!\cdots\!78\)\( p^{13} T^{26} + \)\(67\!\cdots\!54\)\( p^{16} T^{28} - 19025559809330919362 p^{20} T^{30} + 4655644696317798 p^{24} T^{32} - 960261674600 p^{28} T^{34} + 156083133 p^{32} T^{36} - 17518 p^{36} T^{38} + p^{40} T^{40} \)
53 \( ( 1 + 4 T + 15620 T^{2} + 38666 T^{3} + 123697764 T^{4} + 2783688 T^{5} + 647196430990 T^{6} - 1415010838372 T^{7} + 2523792463046985 T^{8} - 8540842263006562 T^{9} + 7846154139849429720 T^{10} - 8540842263006562 p^{2} T^{11} + 2523792463046985 p^{4} T^{12} - 1415010838372 p^{6} T^{13} + 647196430990 p^{8} T^{14} + 2783688 p^{10} T^{15} + 123697764 p^{12} T^{16} + 38666 p^{14} T^{17} + 15620 p^{16} T^{18} + 4 p^{18} T^{19} + p^{20} T^{20} )^{2} \)
61 \( 1 - 29346 T^{2} + 453577409 T^{4} - 4851615536100 T^{6} + 40329899735365142 T^{8} - \)\(27\!\cdots\!70\)\( T^{10} + \)\(16\!\cdots\!54\)\( T^{12} - \)\(84\!\cdots\!82\)\( T^{14} + \)\(39\!\cdots\!77\)\( T^{16} - \)\(16\!\cdots\!66\)\( T^{18} + \)\(64\!\cdots\!06\)\( T^{20} - \)\(16\!\cdots\!66\)\( p^{4} T^{22} + \)\(39\!\cdots\!77\)\( p^{8} T^{24} - \)\(84\!\cdots\!82\)\( p^{12} T^{26} + \)\(16\!\cdots\!54\)\( p^{16} T^{28} - \)\(27\!\cdots\!70\)\( p^{20} T^{30} + 40329899735365142 p^{24} T^{32} - 4851615536100 p^{28} T^{34} + 453577409 p^{32} T^{36} - 29346 p^{36} T^{38} + p^{40} T^{40} \)
67 \( 1 - 29100 T^{2} + 521030128 T^{4} - 6880917032148 T^{6} + 73053607440768962 T^{8} - \)\(65\!\cdots\!48\)\( T^{10} + \)\(49\!\cdots\!90\)\( T^{12} - \)\(33\!\cdots\!72\)\( T^{14} + \)\(19\!\cdots\!73\)\( T^{16} - \)\(10\!\cdots\!04\)\( T^{18} + \)\(50\!\cdots\!88\)\( T^{20} - \)\(10\!\cdots\!04\)\( p^{4} T^{22} + \)\(19\!\cdots\!73\)\( p^{8} T^{24} - \)\(33\!\cdots\!72\)\( p^{12} T^{26} + \)\(49\!\cdots\!90\)\( p^{16} T^{28} - \)\(65\!\cdots\!48\)\( p^{20} T^{30} + 73053607440768962 p^{24} T^{32} - 6880917032148 p^{28} T^{34} + 521030128 p^{32} T^{36} - 29100 p^{36} T^{38} + p^{40} T^{40} \)
71 \( ( 1 - 46 T + 36049 T^{2} - 1508788 T^{3} + 619396726 T^{4} - 23137924648 T^{5} + 6756698698101 T^{6} - 223152723917514 T^{7} + 52333617071040123 T^{8} - 1520457065078166428 T^{9} + \)\(30\!\cdots\!06\)\( T^{10} - 1520457065078166428 p^{2} T^{11} + 52333617071040123 p^{4} T^{12} - 223152723917514 p^{6} T^{13} + 6756698698101 p^{8} T^{14} - 23137924648 p^{10} T^{15} + 619396726 p^{12} T^{16} - 1508788 p^{14} T^{17} + 36049 p^{16} T^{18} - 46 p^{18} T^{19} + p^{20} T^{20} )^{2} \)
73 \( 1 - 44790 T^{2} + 1067281693 T^{4} - 17526901495860 T^{6} + 221031220384497830 T^{8} - \)\(22\!\cdots\!94\)\( T^{10} + \)\(19\!\cdots\!50\)\( T^{12} - \)\(14\!\cdots\!86\)\( T^{14} + \)\(10\!\cdots\!29\)\( T^{16} - \)\(61\!\cdots\!18\)\( T^{18} + \)\(34\!\cdots\!94\)\( T^{20} - \)\(61\!\cdots\!18\)\( p^{4} T^{22} + \)\(10\!\cdots\!29\)\( p^{8} T^{24} - \)\(14\!\cdots\!86\)\( p^{12} T^{26} + \)\(19\!\cdots\!50\)\( p^{16} T^{28} - \)\(22\!\cdots\!94\)\( p^{20} T^{30} + 221031220384497830 p^{24} T^{32} - 17526901495860 p^{28} T^{34} + 1067281693 p^{32} T^{36} - 44790 p^{36} T^{38} + p^{40} T^{40} \)
79 \( ( 1 + 210 T + 55013 T^{2} + 7637262 T^{3} + 1211778452 T^{4} + 130666986398 T^{5} + 15907885114307 T^{6} + 1431668961080462 T^{7} + 146389736312489335 T^{8} + 11461287473659993972 T^{9} + \)\(10\!\cdots\!80\)\( T^{10} + 11461287473659993972 p^{2} T^{11} + 146389736312489335 p^{4} T^{12} + 1431668961080462 p^{6} T^{13} + 15907885114307 p^{8} T^{14} + 130666986398 p^{10} T^{15} + 1211778452 p^{12} T^{16} + 7637262 p^{14} T^{17} + 55013 p^{16} T^{18} + 210 p^{18} T^{19} + p^{20} T^{20} )^{2} \)
83 \( 1 - 77388 T^{2} + 2941221734 T^{4} - 73711040744874 T^{6} + 1376827539396685037 T^{8} - \)\(20\!\cdots\!18\)\( T^{10} + \)\(25\!\cdots\!05\)\( T^{12} - \)\(26\!\cdots\!78\)\( T^{14} + \)\(24\!\cdots\!82\)\( T^{16} - \)\(20\!\cdots\!90\)\( T^{18} + \)\(14\!\cdots\!42\)\( T^{20} - \)\(20\!\cdots\!90\)\( p^{4} T^{22} + \)\(24\!\cdots\!82\)\( p^{8} T^{24} - \)\(26\!\cdots\!78\)\( p^{12} T^{26} + \)\(25\!\cdots\!05\)\( p^{16} T^{28} - \)\(20\!\cdots\!18\)\( p^{20} T^{30} + 1376827539396685037 p^{24} T^{32} - 73711040744874 p^{28} T^{34} + 2941221734 p^{32} T^{36} - 77388 p^{36} T^{38} + p^{40} T^{40} \)
89 \( 1 - 74702 T^{2} + 2706108269 T^{4} - 63436412992128 T^{6} + 1089431865086821382 T^{8} - \)\(14\!\cdots\!30\)\( T^{10} + \)\(16\!\cdots\!42\)\( T^{12} - \)\(16\!\cdots\!22\)\( T^{14} + \)\(14\!\cdots\!77\)\( T^{16} - \)\(12\!\cdots\!82\)\( T^{18} + \)\(98\!\cdots\!42\)\( T^{20} - \)\(12\!\cdots\!82\)\( p^{4} T^{22} + \)\(14\!\cdots\!77\)\( p^{8} T^{24} - \)\(16\!\cdots\!22\)\( p^{12} T^{26} + \)\(16\!\cdots\!42\)\( p^{16} T^{28} - \)\(14\!\cdots\!30\)\( p^{20} T^{30} + 1089431865086821382 p^{24} T^{32} - 63436412992128 p^{28} T^{34} + 2706108269 p^{32} T^{36} - 74702 p^{36} T^{38} + p^{40} T^{40} \)
97 \( 1 - 73436 T^{2} + 3000462704 T^{4} - 86935592221236 T^{6} + 1969111993183071938 T^{8} - \)\(36\!\cdots\!80\)\( T^{10} + \)\(58\!\cdots\!98\)\( T^{12} - \)\(80\!\cdots\!88\)\( T^{14} + \)\(99\!\cdots\!05\)\( T^{16} - \)\(10\!\cdots\!28\)\( T^{18} + \)\(10\!\cdots\!08\)\( T^{20} - \)\(10\!\cdots\!28\)\( p^{4} T^{22} + \)\(99\!\cdots\!05\)\( p^{8} T^{24} - \)\(80\!\cdots\!88\)\( p^{12} T^{26} + \)\(58\!\cdots\!98\)\( p^{16} T^{28} - \)\(36\!\cdots\!80\)\( p^{20} T^{30} + 1969111993183071938 p^{24} T^{32} - 86935592221236 p^{28} T^{34} + 3000462704 p^{32} T^{36} - 73436 p^{36} T^{38} + p^{40} T^{40} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.84453489609604153094783936196, −2.81542931313248654465562691800, −2.74405032590947874509790643102, −2.51659899709513722158726777305, −2.39387978226828740447239763181, −2.39317856441518107229181598514, −2.38060865724437058772573282141, −2.32402830146647060804930381053, −2.10630613753167731574607459577, −2.03545393300132907733747254476, −1.96778098985254429067853212716, −1.94875150719745852764418749014, −1.89412007071514535499698581334, −1.83025235962090972816089448903, −1.73674660453207865961493210556, −1.63729299256276618711374473467, −1.52834207301522552586724285155, −1.38030843357097825901642573443, −1.25948377869023240248073928998, −1.18882402833933497018196481866, −1.17879301064979186212655335437, −0.55100635932318127334360703787, −0.28283939527970848815174374317, −0.17897300128400760021642409161, −0.05994634259757846635793119946, 0.05994634259757846635793119946, 0.17897300128400760021642409161, 0.28283939527970848815174374317, 0.55100635932318127334360703787, 1.17879301064979186212655335437, 1.18882402833933497018196481866, 1.25948377869023240248073928998, 1.38030843357097825901642573443, 1.52834207301522552586724285155, 1.63729299256276618711374473467, 1.73674660453207865961493210556, 1.83025235962090972816089448903, 1.89412007071514535499698581334, 1.94875150719745852764418749014, 1.96778098985254429067853212716, 2.03545393300132907733747254476, 2.10630613753167731574607459577, 2.32402830146647060804930381053, 2.38060865724437058772573282141, 2.39317856441518107229181598514, 2.39387978226828740447239763181, 2.51659899709513722158726777305, 2.74405032590947874509790643102, 2.81542931313248654465562691800, 2.84453489609604153094783936196

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.