Properties

Label 20-1008e10-1.1-c1e10-0-0
Degree $20$
Conductor $1.083\times 10^{30}$
Sign $1$
Analytic cond. $1.14123\times 10^{9}$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 5·7-s − 4·11-s − 3·13-s − 2·19-s − 8·23-s + 12·25-s + 3·27-s − 9·29-s + 3·31-s + 15·35-s − 6·37-s − 12·41-s + 5·43-s − 3·47-s + 10·49-s + 60·53-s + 12·55-s − 7·59-s − 14·61-s + 9·65-s + 8·67-s + 18·71-s + 30·73-s + 20·77-s + 3·79-s − 3·81-s + ⋯
L(s)  = 1  − 1.34·5-s − 1.88·7-s − 1.20·11-s − 0.832·13-s − 0.458·19-s − 1.66·23-s + 12/5·25-s + 0.577·27-s − 1.67·29-s + 0.538·31-s + 2.53·35-s − 0.986·37-s − 1.87·41-s + 0.762·43-s − 0.437·47-s + 10/7·49-s + 8.24·53-s + 1.61·55-s − 0.911·59-s − 1.79·61-s + 1.11·65-s + 0.977·67-s + 2.13·71-s + 3.51·73-s + 2.27·77-s + 0.337·79-s − 1/3·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{40} \cdot 3^{20} \cdot 7^{10}\)
Sign: $1$
Analytic conductor: \(1.14123\times 10^{9}\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{40} \cdot 3^{20} \cdot 7^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5050614781\)
\(L(\frac12)\) \(\approx\) \(0.5050614781\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - p T^{3} + p T^{4} + 2 p^{2} T^{5} + p^{2} T^{6} - p^{3} T^{7} + p^{5} T^{10} \)
7 \( ( 1 + T + T^{2} )^{5} \)
good5 \( 1 + 3 T - 3 T^{2} - 16 T^{3} - p^{2} T^{4} - 63 T^{5} - 19 T^{6} + 277 T^{7} + 31 p^{2} T^{8} + 451 T^{9} - 1814 T^{10} + 451 p T^{11} + 31 p^{4} T^{12} + 277 p^{3} T^{13} - 19 p^{4} T^{14} - 63 p^{5} T^{15} - p^{8} T^{16} - 16 p^{7} T^{17} - 3 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \)
11 \( 1 + 4 T - 6 T^{2} + 10 T^{3} + 191 T^{4} - 240 T^{5} + 139 p T^{6} + 16178 T^{7} - 5837 T^{8} - 14002 T^{9} + 502605 T^{10} - 14002 p T^{11} - 5837 p^{2} T^{12} + 16178 p^{3} T^{13} + 139 p^{5} T^{14} - 240 p^{5} T^{15} + 191 p^{6} T^{16} + 10 p^{7} T^{17} - 6 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \)
13 \( 1 + 3 T - 31 T^{2} - 106 T^{3} + 402 T^{4} + 1426 T^{5} - 5437 T^{6} - 15135 T^{7} + 85123 T^{8} + 101952 T^{9} - 1104756 T^{10} + 101952 p T^{11} + 85123 p^{2} T^{12} - 15135 p^{3} T^{13} - 5437 p^{4} T^{14} + 1426 p^{5} T^{15} + 402 p^{6} T^{16} - 106 p^{7} T^{17} - 31 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \)
17 \( ( 1 + 46 T^{2} - 9 T^{3} + 1261 T^{4} - 198 T^{5} + 1261 p T^{6} - 9 p^{2} T^{7} + 46 p^{3} T^{8} + p^{5} T^{10} )^{2} \)
19 \( ( 1 + T + 42 T^{2} + 66 T^{3} + 1281 T^{4} + 1209 T^{5} + 1281 p T^{6} + 66 p^{2} T^{7} + 42 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \)
23 \( 1 + 8 T + 9 T^{2} + 222 T^{3} + 2650 T^{4} + 9138 T^{5} + 54826 T^{6} + 441030 T^{7} + 71882 p T^{8} + 8929410 T^{9} + 60424408 T^{10} + 8929410 p T^{11} + 71882 p^{3} T^{12} + 441030 p^{3} T^{13} + 54826 p^{4} T^{14} + 9138 p^{5} T^{15} + 2650 p^{6} T^{16} + 222 p^{7} T^{17} + 9 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \)
29 \( 1 + 9 T - 52 T^{2} - 657 T^{3} + 1944 T^{4} + 29007 T^{5} - 46389 T^{6} - 788049 T^{7} + 920775 T^{8} + 10497078 T^{9} - 9302769 T^{10} + 10497078 p T^{11} + 920775 p^{2} T^{12} - 788049 p^{3} T^{13} - 46389 p^{4} T^{14} + 29007 p^{5} T^{15} + 1944 p^{6} T^{16} - 657 p^{7} T^{17} - 52 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \)
31 \( 1 - 3 T - 106 T^{2} + 157 T^{3} + 6498 T^{4} - 2989 T^{5} - 283525 T^{6} - 117 p T^{7} + 9910699 T^{8} + 942522 T^{9} - 316500783 T^{10} + 942522 p T^{11} + 9910699 p^{2} T^{12} - 117 p^{4} T^{13} - 283525 p^{4} T^{14} - 2989 p^{5} T^{15} + 6498 p^{6} T^{16} + 157 p^{7} T^{17} - 106 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
37 \( ( 1 + 3 T + 103 T^{2} + 197 T^{3} + 6112 T^{4} + 11312 T^{5} + 6112 p T^{6} + 197 p^{2} T^{7} + 103 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
41 \( 1 + 12 T - 30 T^{2} - 790 T^{3} - 277 T^{4} + 13968 T^{5} - 140563 T^{6} - 610562 T^{7} + 9173539 T^{8} + 24360490 T^{9} - 273097535 T^{10} + 24360490 p T^{11} + 9173539 p^{2} T^{12} - 610562 p^{3} T^{13} - 140563 p^{4} T^{14} + 13968 p^{5} T^{15} - 277 p^{6} T^{16} - 790 p^{7} T^{17} - 30 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \)
43 \( 1 - 5 T - 88 T^{2} - 275 T^{3} + 6540 T^{4} + 38663 T^{5} - 127543 T^{6} - 2251353 T^{7} - 2752001 T^{8} + 643014 p T^{9} + 456453449 T^{10} + 643014 p^{2} T^{11} - 2752001 p^{2} T^{12} - 2251353 p^{3} T^{13} - 127543 p^{4} T^{14} + 38663 p^{5} T^{15} + 6540 p^{6} T^{16} - 275 p^{7} T^{17} - 88 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \)
47 \( 1 + 3 T - 24 T^{2} - 751 T^{3} - 2770 T^{4} + 11169 T^{5} + 300305 T^{6} + 1566601 T^{7} - 499337 T^{8} - 58566752 T^{9} - 599985995 T^{10} - 58566752 p T^{11} - 499337 p^{2} T^{12} + 1566601 p^{3} T^{13} + 300305 p^{4} T^{14} + 11169 p^{5} T^{15} - 2770 p^{6} T^{16} - 751 p^{7} T^{17} - 24 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \)
53 \( ( 1 - 30 T + 580 T^{2} - 7683 T^{3} + 79903 T^{4} - 646182 T^{5} + 79903 p T^{6} - 7683 p^{2} T^{7} + 580 p^{3} T^{8} - 30 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
59 \( 1 + 7 T - 84 T^{2} - 891 T^{3} - 548 T^{4} + 21051 T^{5} + 190525 T^{6} + 1336983 T^{7} + 78221 p T^{8} - 49687002 T^{9} - 884146943 T^{10} - 49687002 p T^{11} + 78221 p^{3} T^{12} + 1336983 p^{3} T^{13} + 190525 p^{4} T^{14} + 21051 p^{5} T^{15} - 548 p^{6} T^{16} - 891 p^{7} T^{17} - 84 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \)
61 \( 1 + 14 T - 67 T^{2} - 1032 T^{3} + 10040 T^{4} + 70122 T^{5} - 675608 T^{6} - 3001836 T^{7} + 20748698 T^{8} - 6195900 T^{9} - 1395441908 T^{10} - 6195900 p T^{11} + 20748698 p^{2} T^{12} - 3001836 p^{3} T^{13} - 675608 p^{4} T^{14} + 70122 p^{5} T^{15} + 10040 p^{6} T^{16} - 1032 p^{7} T^{17} - 67 p^{8} T^{18} + 14 p^{9} T^{19} + p^{10} T^{20} \)
67 \( 1 - 8 T - 142 T^{2} + 1462 T^{3} + 7779 T^{4} - 110116 T^{5} - 343663 T^{6} + 6410154 T^{7} + 10809007 T^{8} - 209041554 T^{9} + 61061861 T^{10} - 209041554 p T^{11} + 10809007 p^{2} T^{12} + 6410154 p^{3} T^{13} - 343663 p^{4} T^{14} - 110116 p^{5} T^{15} + 7779 p^{6} T^{16} + 1462 p^{7} T^{17} - 142 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \)
71 \( ( 1 - 9 T + 306 T^{2} - 2174 T^{3} + 40411 T^{4} - 221049 T^{5} + 40411 p T^{6} - 2174 p^{2} T^{7} + 306 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
73 \( ( 1 - 15 T + 247 T^{2} - 2719 T^{3} + 26596 T^{4} - 243028 T^{5} + 26596 p T^{6} - 2719 p^{2} T^{7} + 247 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
79 \( 1 - 3 T - 271 T^{2} + 1532 T^{3} + 38715 T^{4} - 263753 T^{5} - 3446203 T^{6} + 24313053 T^{7} + 236347015 T^{8} - 845708523 T^{9} - 15972358530 T^{10} - 845708523 p T^{11} + 236347015 p^{2} T^{12} + 24313053 p^{3} T^{13} - 3446203 p^{4} T^{14} - 263753 p^{5} T^{15} + 38715 p^{6} T^{16} + 1532 p^{7} T^{17} - 271 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \)
83 \( 1 + 20 T + 98 T^{2} + 318 T^{3} + 879 T^{4} - 169800 T^{5} - 2124519 T^{6} - 6011178 T^{7} + 51057123 T^{8} + 1398046658 T^{9} + 18343755493 T^{10} + 1398046658 p T^{11} + 51057123 p^{2} T^{12} - 6011178 p^{3} T^{13} - 2124519 p^{4} T^{14} - 169800 p^{5} T^{15} + 879 p^{6} T^{16} + 318 p^{7} T^{17} + 98 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} \)
89 \( ( 1 + 12 T + 292 T^{2} + 2463 T^{3} + 41383 T^{4} + 291726 T^{5} + 41383 p T^{6} + 2463 p^{2} T^{7} + 292 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
97 \( 1 + 37 T + 608 T^{2} + 4491 T^{3} - 11038 T^{4} - 630495 T^{5} - 6947105 T^{6} - 31739955 T^{7} + 128335391 T^{8} + 3636093960 T^{9} + 40545974467 T^{10} + 3636093960 p T^{11} + 128335391 p^{2} T^{12} - 31739955 p^{3} T^{13} - 6947105 p^{4} T^{14} - 630495 p^{5} T^{15} - 11038 p^{6} T^{16} + 4491 p^{7} T^{17} + 608 p^{8} T^{18} + 37 p^{9} T^{19} + p^{10} T^{20} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.57814351258623058632719662901, −3.57231303561639118980772222792, −3.47049692539948596375062337018, −3.38441544519663050975135861115, −3.23969792929613562036046739164, −3.06764958013257817949444250087, −2.87140613458151273876715817291, −2.82263934107484695610979056950, −2.76237513364160229980909953749, −2.70212005411369004476048397917, −2.54122202383991343167595829485, −2.46145568926162256221890669089, −2.06413228559660619548143299295, −2.05296951908266413716607721219, −2.03654979697388932407349516847, −1.97005257851253576978912194307, −1.92786658026348059706303638412, −1.60518462862470276107055490702, −1.22861206956595912424795584311, −1.14937990872457129612123316493, −0.817942560789594847978793424786, −0.74654044867080083089052220714, −0.70805006013412163898916553187, −0.27623428029690580170783483358, −0.15397476752863631383426890477, 0.15397476752863631383426890477, 0.27623428029690580170783483358, 0.70805006013412163898916553187, 0.74654044867080083089052220714, 0.817942560789594847978793424786, 1.14937990872457129612123316493, 1.22861206956595912424795584311, 1.60518462862470276107055490702, 1.92786658026348059706303638412, 1.97005257851253576978912194307, 2.03654979697388932407349516847, 2.05296951908266413716607721219, 2.06413228559660619548143299295, 2.46145568926162256221890669089, 2.54122202383991343167595829485, 2.70212005411369004476048397917, 2.76237513364160229980909953749, 2.82263934107484695610979056950, 2.87140613458151273876715817291, 3.06764958013257817949444250087, 3.23969792929613562036046739164, 3.38441544519663050975135861115, 3.47049692539948596375062337018, 3.57231303561639118980772222792, 3.57814351258623058632719662901

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

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