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Properties

Label 20-287e10-1.1-c1e10-0-0
Degree $20$
Conductor $3.792\times 10^{24}$
Sign $1$
Analytic cond. $3995.67$
Root an. cond. $1.51383$
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  + 4·2-s + 4·4-s − 10·5-s − 4·8-s + 10·9-s − 40·10-s − 18·16-s + 40·18-s − 40·20-s − 4·23-s + 23·25-s + 14·31-s − 38·32-s + 40·36-s − 22·37-s + 40·40-s − 4·41-s + 18·43-s − 100·45-s − 16·46-s − 5·49-s + 92·50-s + 34·59-s − 28·61-s + 56·62-s − 28·64-s − 40·72-s + ⋯
L(s)  = 1  + 2.82·2-s + 2·4-s − 4.47·5-s − 1.41·8-s + 10/3·9-s − 12.6·10-s − 9/2·16-s + 9.42·18-s − 8.94·20-s − 0.834·23-s + 23/5·25-s + 2.51·31-s − 6.71·32-s + 20/3·36-s − 3.61·37-s + 6.32·40-s − 0.624·41-s + 2.74·43-s − 14.9·45-s − 2.35·46-s − 5/7·49-s + 13.0·50-s + 4.42·59-s − 3.58·61-s + 7.11·62-s − 7/2·64-s − 4.71·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 41^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 41^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(7^{10} \cdot 41^{10}\)
Sign: $1$
Analytic conductor: \(3995.67\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 7^{10} \cdot 41^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.09195559097\)
\(L(\frac12)\) \(\approx\) \(0.09195559097\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( ( 1 + T^{2} )^{5} \)
41 \( 1 + 4 T - 47 T^{2} + 144 T^{3} + 1942 T^{4} - 3688 T^{5} + 1942 p T^{6} + 144 p^{2} T^{7} - 47 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
good2 \( ( 1 - p T + p^{2} T^{2} - 3 p T^{3} + 13 T^{4} - 19 T^{5} + 13 p T^{6} - 3 p^{3} T^{7} + p^{5} T^{8} - p^{5} T^{9} + p^{5} T^{10} )^{2} \)
3 \( 1 - 10 T^{2} + 55 T^{4} - 224 T^{6} + 769 T^{8} - 2417 T^{10} + 769 p^{2} T^{12} - 224 p^{4} T^{14} + 55 p^{6} T^{16} - 10 p^{8} T^{18} + p^{10} T^{20} \)
5 \( ( 1 + p T + 26 T^{2} + 84 T^{3} + 261 T^{4} + 598 T^{5} + 261 p T^{6} + 84 p^{2} T^{7} + 26 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} )^{2} \)
11 \( 1 - 64 T^{2} + 2166 T^{4} - 49150 T^{6} + 814601 T^{8} - 10239268 T^{10} + 814601 p^{2} T^{12} - 49150 p^{4} T^{14} + 2166 p^{6} T^{16} - 64 p^{8} T^{18} + p^{10} T^{20} \)
13 \( 1 - 59 T^{2} + 1942 T^{4} - 42976 T^{6} + 737713 T^{8} - 10374425 T^{10} + 737713 p^{2} T^{12} - 42976 p^{4} T^{14} + 1942 p^{6} T^{16} - 59 p^{8} T^{18} + p^{10} T^{20} \)
17 \( 1 - 115 T^{2} + 6566 T^{4} - 242720 T^{6} + 6399649 T^{8} - 7378185 p T^{10} + 6399649 p^{2} T^{12} - 242720 p^{4} T^{14} + 6566 p^{6} T^{16} - 115 p^{8} T^{18} + p^{10} T^{20} \)
19 \( 1 - 94 T^{2} + 4603 T^{4} - 151512 T^{6} + 3820801 T^{8} - 79003953 T^{10} + 3820801 p^{2} T^{12} - 151512 p^{4} T^{14} + 4603 p^{6} T^{16} - 94 p^{8} T^{18} + p^{10} T^{20} \)
23 \( ( 1 + 2 T + 59 T^{2} + 176 T^{3} + 2067 T^{4} + 5357 T^{5} + 2067 p T^{6} + 176 p^{2} T^{7} + 59 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
29 \( 1 - 91 T^{2} + 5518 T^{4} - 214978 T^{6} + 7185025 T^{8} - 204873190 T^{10} + 7185025 p^{2} T^{12} - 214978 p^{4} T^{14} + 5518 p^{6} T^{16} - 91 p^{8} T^{18} + p^{10} T^{20} \)
31 \( ( 1 - 7 T + 104 T^{2} - 586 T^{3} + 5279 T^{4} - 22742 T^{5} + 5279 p T^{6} - 586 p^{2} T^{7} + 104 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
37 \( ( 1 + 11 T + 129 T^{2} + 1051 T^{3} + 8808 T^{4} + 52380 T^{5} + 8808 p T^{6} + 1051 p^{2} T^{7} + 129 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
43 \( ( 1 - 9 T + 150 T^{2} - 1070 T^{3} + 10507 T^{4} - 58949 T^{5} + 10507 p T^{6} - 1070 p^{2} T^{7} + 150 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
47 \( 1 - 205 T^{2} + 17547 T^{4} - 626815 T^{6} - 7566568 T^{8} + 1397327840 T^{10} - 7566568 p^{2} T^{12} - 626815 p^{4} T^{14} + 17547 p^{6} T^{16} - 205 p^{8} T^{18} + p^{10} T^{20} \)
53 \( 1 - 235 T^{2} + 27554 T^{4} - 2210254 T^{6} + 143685261 T^{8} - 8113437742 T^{10} + 143685261 p^{2} T^{12} - 2210254 p^{4} T^{14} + 27554 p^{6} T^{16} - 235 p^{8} T^{18} + p^{10} T^{20} \)
59 \( ( 1 - 17 T + 144 T^{2} - 1134 T^{3} + 11155 T^{4} - 94458 T^{5} + 11155 p T^{6} - 1134 p^{2} T^{7} + 144 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
61 \( ( 1 + 14 T + 314 T^{2} + 3100 T^{3} + 38149 T^{4} + 273684 T^{5} + 38149 p T^{6} + 3100 p^{2} T^{7} + 314 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
67 \( 1 - 427 T^{2} + 89738 T^{4} - 12380986 T^{6} + 1246484565 T^{8} - 95340189814 T^{10} + 1246484565 p^{2} T^{12} - 12380986 p^{4} T^{14} + 89738 p^{6} T^{16} - 427 p^{8} T^{18} + p^{10} T^{20} \)
71 \( 1 - 264 T^{2} + 49006 T^{4} - 5988070 T^{6} + 599427089 T^{8} - 46178547364 T^{10} + 599427089 p^{2} T^{12} - 5988070 p^{4} T^{14} + 49006 p^{6} T^{16} - 264 p^{8} T^{18} + p^{10} T^{20} \)
73 \( ( 1 + 210 T^{2} - 834 T^{3} + 18641 T^{4} - 121708 T^{5} + 18641 p T^{6} - 834 p^{2} T^{7} + 210 p^{3} T^{8} + p^{5} T^{10} )^{2} \)
79 \( 1 - 18 T^{2} + 20301 T^{4} - 484504 T^{6} + 185399314 T^{8} - 4645230508 T^{10} + 185399314 p^{2} T^{12} - 484504 p^{4} T^{14} + 20301 p^{6} T^{16} - 18 p^{8} T^{18} + p^{10} T^{20} \)
83 \( ( 1 + 10 T + 288 T^{2} + 2608 T^{3} + 42487 T^{4} + 296724 T^{5} + 42487 p T^{6} + 2608 p^{2} T^{7} + 288 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
89 \( 1 - 454 T^{2} + 89791 T^{4} - 10376100 T^{6} + 846113269 T^{8} - 67027887669 T^{10} + 846113269 p^{2} T^{12} - 10376100 p^{4} T^{14} + 89791 p^{6} T^{16} - 454 p^{8} T^{18} + p^{10} T^{20} \)
97 \( 1 - 478 T^{2} + 119351 T^{4} - 20862580 T^{6} + 2835760101 T^{8} - 307720470589 T^{10} + 2835760101 p^{2} T^{12} - 20862580 p^{4} T^{14} + 119351 p^{6} T^{16} - 478 p^{8} T^{18} + p^{10} T^{20} \)
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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

−4.59855189632102450952451168001, −4.27609708286718203543363990741, −4.26693285632296156015569300754, −4.20313398631127862755777917570, −4.17991434253598939917359518330, −4.14111064048756927960026776729, −3.92271515481365369067871602161, −3.80847280530696687349302766503, −3.65124711685104607007899212785, −3.59002888529390789412536470625, −3.55778627543154624281618625286, −3.35854959699572536014760803448, −3.31688770921343737159556408975, −2.85335289847605660266476895140, −2.83435754709374218739923272395, −2.76750286080905500384408081578, −2.25885845374327437975513660765, −2.08889081063273161800049071327, −2.05274183957637519439523296982, −1.85967889635050757125504366367, −1.68772725939601050888995456510, −1.40105426968644163854362106121, −0.940100185991800980986837497682, −0.60575748161309596718425625909, −0.06579849058282776868923995970, 0.06579849058282776868923995970, 0.60575748161309596718425625909, 0.940100185991800980986837497682, 1.40105426968644163854362106121, 1.68772725939601050888995456510, 1.85967889635050757125504366367, 2.05274183957637519439523296982, 2.08889081063273161800049071327, 2.25885845374327437975513660765, 2.76750286080905500384408081578, 2.83435754709374218739923272395, 2.85335289847605660266476895140, 3.31688770921343737159556408975, 3.35854959699572536014760803448, 3.55778627543154624281618625286, 3.59002888529390789412536470625, 3.65124711685104607007899212785, 3.80847280530696687349302766503, 3.92271515481365369067871602161, 4.14111064048756927960026776729, 4.17991434253598939917359518330, 4.20313398631127862755777917570, 4.26693285632296156015569300754, 4.27609708286718203543363990741, 4.59855189632102450952451168001

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

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