Properties

Label 40-605e20-1.1-c1e20-0-0
Degree $40$
Conductor $4.316\times 10^{55}$
Sign $1$
Analytic cond. $4.79346\times 10^{13}$
Root an. cond. $2.19794$
Motivic weight $1$
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  + 4·3-s + 4·5-s + 8·9-s + 16·15-s + 13·16-s + 12·23-s + 16·25-s + 8·27-s − 16·31-s − 72·37-s + 32·45-s + 16·47-s + 52·48-s − 52·53-s + 28·67-s + 48·69-s + 24·71-s + 64·75-s + 52·80-s + 40·81-s − 64·93-s + 32·97-s − 72·103-s − 288·111-s − 88·113-s + 48·115-s + 68·125-s + ⋯
L(s)  = 1  + 2.30·3-s + 1.78·5-s + 8/3·9-s + 4.13·15-s + 13/4·16-s + 2.50·23-s + 16/5·25-s + 1.53·27-s − 2.87·31-s − 11.8·37-s + 4.77·45-s + 2.33·47-s + 7.50·48-s − 7.14·53-s + 3.42·67-s + 5.77·69-s + 2.84·71-s + 7.39·75-s + 5.81·80-s + 40/9·81-s − 6.63·93-s + 3.24·97-s − 7.09·103-s − 27.3·111-s − 8.27·113-s + 4.47·115-s + 6.08·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 11^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 11^{40}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(5^{20} \cdot 11^{40}\)
Sign: $1$
Analytic conductor: \(4.79346\times 10^{13}\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 5^{20} \cdot 11^{40} ,\ ( \ : [1/2]^{20} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.823466578\)
\(L(\frac12)\) \(\approx\) \(7.823466578\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 - 2 T - 2 T^{2} - 6 T^{3} - 7 T^{4} + 128 T^{5} - 7 p T^{6} - 6 p^{2} T^{7} - 2 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
11 \( 1 \)
good2 \( 1 - 13 T^{4} + 67 T^{8} - 103 T^{12} - 111 p^{3} T^{16} + 403 p^{4} T^{20} - 111 p^{7} T^{24} - 103 p^{8} T^{28} + 67 p^{12} T^{32} - 13 p^{16} T^{36} + p^{20} T^{40} \)
3 \( ( 1 - 2 T + 2 T^{2} - 22 T^{4} + 10 p T^{5} - 16 T^{6} - 2 p^{2} T^{7} + 121 T^{8} - 70 T^{9} - 4 T^{10} - 70 p T^{11} + 121 p^{2} T^{12} - 2 p^{5} T^{13} - 16 p^{4} T^{14} + 10 p^{6} T^{15} - 22 p^{6} T^{16} + 2 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
7 \( 1 - 62 T^{4} + 5469 T^{8} + 23544 T^{12} - 475282 p T^{16} + 865653644 T^{20} - 475282 p^{5} T^{24} + 23544 p^{8} T^{28} + 5469 p^{12} T^{32} - 62 p^{16} T^{36} + p^{20} T^{40} \)
13 \( 1 - 486 T^{4} + 137533 T^{8} - 33338824 T^{12} + 6515163730 T^{16} - 1105389843492 T^{20} + 6515163730 p^{4} T^{24} - 33338824 p^{8} T^{28} + 137533 p^{12} T^{32} - 486 p^{16} T^{36} + p^{20} T^{40} \)
17 \( 1 + 1154 T^{4} + 603453 T^{8} + 206648568 T^{12} + 61024612610 T^{16} + 17793790086924 T^{20} + 61024612610 p^{4} T^{24} + 206648568 p^{8} T^{28} + 603453 p^{12} T^{32} + 1154 p^{16} T^{36} + p^{20} T^{40} \)
19 \( ( 1 + 102 T^{2} + 5637 T^{4} + 210984 T^{6} + 5868002 T^{8} + 126100900 T^{10} + 5868002 p^{2} T^{12} + 210984 p^{4} T^{14} + 5637 p^{6} T^{16} + 102 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
23 \( ( 1 - 6 T + 18 T^{2} - 88 T^{3} - 390 T^{4} + 3642 T^{5} - 10960 T^{6} + 59522 T^{7} - 18775 T^{8} - 727466 T^{9} + 1822492 T^{10} - 727466 p T^{11} - 18775 p^{2} T^{12} + 59522 p^{3} T^{13} - 10960 p^{4} T^{14} + 3642 p^{5} T^{15} - 390 p^{6} T^{16} - 88 p^{7} T^{17} + 18 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
29 \( ( 1 + 188 T^{2} + 17345 T^{4} + 1040848 T^{6} + 45231494 T^{8} + 1493813160 T^{10} + 45231494 p^{2} T^{12} + 1040848 p^{4} T^{14} + 17345 p^{6} T^{16} + 188 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
31 \( ( 1 + 4 T + 96 T^{2} + 196 T^{3} + 4427 T^{4} + 6736 T^{5} + 4427 p T^{6} + 196 p^{2} T^{7} + 96 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{4} \)
37 \( ( 1 + 36 T + 648 T^{2} + 7882 T^{3} + 72702 T^{4} + 538980 T^{5} + 3355346 T^{6} + 18144776 T^{7} + 87934401 T^{8} + 413286606 T^{9} + 2223305708 T^{10} + 413286606 p T^{11} + 87934401 p^{2} T^{12} + 18144776 p^{3} T^{13} + 3355346 p^{4} T^{14} + 538980 p^{5} T^{15} + 72702 p^{6} T^{16} + 7882 p^{7} T^{17} + 648 p^{8} T^{18} + 36 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
41 \( ( 1 - 4 p T^{2} + 15593 T^{4} - 1033424 T^{6} + 54334406 T^{8} - 2394710872 T^{10} + 54334406 p^{2} T^{12} - 1033424 p^{4} T^{14} + 15593 p^{6} T^{16} - 4 p^{9} T^{18} + p^{10} T^{20} )^{2} \)
43 \( 1 - 366 T^{4} + 7198573 T^{8} + 4199255608 T^{12} + 33982553505090 T^{16} + 16528564405822444 T^{20} + 33982553505090 p^{4} T^{24} + 4199255608 p^{8} T^{28} + 7198573 p^{12} T^{32} - 366 p^{16} T^{36} + p^{20} T^{40} \)
47 \( ( 1 - 8 T + 32 T^{2} - 696 T^{3} + 5345 T^{4} + 1712 T^{5} + 57472 T^{6} - 729616 T^{7} - 14228162 T^{8} + 93413648 T^{9} - 89917632 T^{10} + 93413648 p T^{11} - 14228162 p^{2} T^{12} - 729616 p^{3} T^{13} + 57472 p^{4} T^{14} + 1712 p^{5} T^{15} + 5345 p^{6} T^{16} - 696 p^{7} T^{17} + 32 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
53 \( ( 1 + 26 T + 338 T^{2} + 3778 T^{3} + 44197 T^{4} + 456344 T^{5} + 4063000 T^{6} + 35985080 T^{7} + 314014546 T^{8} + 2431350332 T^{9} + 17582336492 T^{10} + 2431350332 p T^{11} + 314014546 p^{2} T^{12} + 35985080 p^{3} T^{13} + 4063000 p^{4} T^{14} + 456344 p^{5} T^{15} + 44197 p^{6} T^{16} + 3778 p^{7} T^{17} + 338 p^{8} T^{18} + 26 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
59 \( ( 1 - 224 T^{2} + 26278 T^{4} - 2420818 T^{6} + 185006377 T^{8} - 11769778460 T^{10} + 185006377 p^{2} T^{12} - 2420818 p^{4} T^{14} + 26278 p^{6} T^{16} - 224 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
61 \( ( 1 - 316 T^{2} + 49425 T^{4} - 5376912 T^{6} + 459195446 T^{8} - 31382550056 T^{10} + 459195446 p^{2} T^{12} - 5376912 p^{4} T^{14} + 49425 p^{6} T^{16} - 316 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
67 \( ( 1 - 14 T + 98 T^{2} - 884 T^{3} + 13938 T^{4} - 145718 T^{5} + 1064856 T^{6} - 10323198 T^{7} + 110769297 T^{8} - 855066006 T^{9} + 6016144844 T^{10} - 855066006 p T^{11} + 110769297 p^{2} T^{12} - 10323198 p^{3} T^{13} + 1064856 p^{4} T^{14} - 145718 p^{5} T^{15} + 13938 p^{6} T^{16} - 884 p^{7} T^{17} + 98 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
71 \( ( 1 - 6 T + 272 T^{2} - 1512 T^{3} + 33803 T^{4} - 156356 T^{5} + 33803 p T^{6} - 1512 p^{2} T^{7} + 272 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} )^{4} \)
73 \( 1 - 19894 T^{4} + 218265741 T^{8} - 1644948134664 T^{12} + 9945143414607122 T^{16} - 54329647222797982596 T^{20} + 9945143414607122 p^{4} T^{24} - 1644948134664 p^{8} T^{28} + 218265741 p^{12} T^{32} - 19894 p^{16} T^{36} + p^{20} T^{40} \)
79 \( ( 1 + 354 T^{2} + 71761 T^{4} + 10348288 T^{6} + 1150587086 T^{8} + 101567433148 T^{10} + 1150587086 p^{2} T^{12} + 10348288 p^{4} T^{14} + 71761 p^{6} T^{16} + 354 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
83 \( 1 + 16242 T^{4} + 248681805 T^{8} + 2697827568568 T^{12} + 23846724347534530 T^{16} + \)\(18\!\cdots\!28\)\( T^{20} + 23846724347534530 p^{4} T^{24} + 2697827568568 p^{8} T^{28} + 248681805 p^{12} T^{32} + 16242 p^{16} T^{36} + p^{20} T^{40} \)
89 \( ( 1 - 608 T^{2} + 175318 T^{4} - 32239126 T^{6} + 4270101689 T^{8} - 431403182580 T^{10} + 4270101689 p^{2} T^{12} - 32239126 p^{4} T^{14} + 175318 p^{6} T^{16} - 608 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
97 \( ( 1 - 16 T + 128 T^{2} - 1778 T^{3} + 25566 T^{4} - 196512 T^{5} + 1452386 T^{6} - 19864876 T^{7} + 376778849 T^{8} - 3580026482 T^{9} + 26549263164 T^{10} - 3580026482 p T^{11} + 376778849 p^{2} T^{12} - 19864876 p^{3} T^{13} + 1452386 p^{4} T^{14} - 196512 p^{5} T^{15} + 25566 p^{6} T^{16} - 1778 p^{7} T^{17} + 128 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} )^{2} \)
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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

−2.49412229903935234963891138306, −2.40840404680173878912394574862, −2.29674289033904330776024682419, −2.07522374802506629226884000019, −2.06699514440091358641131429799, −2.02278716144581011276676247807, −1.97032663658091326974479005666, −1.96588820693278253684997314423, −1.93984552360866815712889291885, −1.88324336037111510339235161980, −1.82050091757370951479176549810, −1.64985936445738826796477192725, −1.58166853337697370510661307253, −1.53504489956119838241180561194, −1.33015749043151101339799738124, −1.30091281473226598347379608668, −1.21416220074531409278478425488, −1.15899467005400552062886140072, −1.08947455329126812026733651196, −1.06832357382458605072370777604, −0.935021467078295567414945220853, −0.75173023845488469791095698002, −0.27236692183794011117527977129, −0.23723155181433295796879750451, −0.16126521349347227318033126977, 0.16126521349347227318033126977, 0.23723155181433295796879750451, 0.27236692183794011117527977129, 0.75173023845488469791095698002, 0.935021467078295567414945220853, 1.06832357382458605072370777604, 1.08947455329126812026733651196, 1.15899467005400552062886140072, 1.21416220074531409278478425488, 1.30091281473226598347379608668, 1.33015749043151101339799738124, 1.53504489956119838241180561194, 1.58166853337697370510661307253, 1.64985936445738826796477192725, 1.82050091757370951479176549810, 1.88324336037111510339235161980, 1.93984552360866815712889291885, 1.96588820693278253684997314423, 1.97032663658091326974479005666, 2.02278716144581011276676247807, 2.06699514440091358641131429799, 2.07522374802506629226884000019, 2.29674289033904330776024682419, 2.40840404680173878912394574862, 2.49412229903935234963891138306

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

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