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Properties

Label 2-507-1.1-c3-0-17
Degree 22
Conductor 507507
Sign 11
Analytic cond. 29.913929.9139
Root an. cond. 5.469365.46936
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 8·4-s + 12·5-s − 2·7-s + 9·9-s + 36·11-s + 24·12-s − 36·15-s + 64·16-s − 78·17-s − 74·19-s − 96·20-s + 6·21-s − 96·23-s + 19·25-s − 27·27-s + 16·28-s + 18·29-s + 214·31-s − 108·33-s − 24·35-s − 72·36-s + 286·37-s + 384·41-s + 524·43-s − 288·44-s + 108·45-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 1.07·5-s − 0.107·7-s + 1/3·9-s + 0.986·11-s + 0.577·12-s − 0.619·15-s + 16-s − 1.11·17-s − 0.893·19-s − 1.07·20-s + 0.0623·21-s − 0.870·23-s + 0.151·25-s − 0.192·27-s + 0.107·28-s + 0.115·29-s + 1.23·31-s − 0.569·33-s − 0.115·35-s − 1/3·36-s + 1.27·37-s + 1.46·41-s + 1.85·43-s − 0.986·44-s + 0.357·45-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: 11
Analytic conductor: 29.913929.9139
Root analytic conductor: 5.469365.46936
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 507, ( :3/2), 1)(2,\ 507,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.4257784131.425778413
L(12)L(\frac12) ≈\approx 1.4257784131.425778413
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+pT 1 + p T
13 1 1
good2 1+p3T2 1 + p^{3} T^{2}
5 1−12T+p3T2 1 - 12 T + p^{3} T^{2}
7 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
11 1−36T+p3T2 1 - 36 T + p^{3} T^{2}
17 1+78T+p3T2 1 + 78 T + p^{3} T^{2}
19 1+74T+p3T2 1 + 74 T + p^{3} T^{2}
23 1+96T+p3T2 1 + 96 T + p^{3} T^{2}
29 1−18T+p3T2 1 - 18 T + p^{3} T^{2}
31 1−214T+p3T2 1 - 214 T + p^{3} T^{2}
37 1−286T+p3T2 1 - 286 T + p^{3} T^{2}
41 1−384T+p3T2 1 - 384 T + p^{3} T^{2}
43 1−524T+p3T2 1 - 524 T + p^{3} T^{2}
47 1+300T+p3T2 1 + 300 T + p^{3} T^{2}
53 1−558T+p3T2 1 - 558 T + p^{3} T^{2}
59 1+576T+p3T2 1 + 576 T + p^{3} T^{2}
61 1−74T+p3T2 1 - 74 T + p^{3} T^{2}
67 1+38T+p3T2 1 + 38 T + p^{3} T^{2}
71 1−456T+p3T2 1 - 456 T + p^{3} T^{2}
73 1−682T+p3T2 1 - 682 T + p^{3} T^{2}
79 1−704T+p3T2 1 - 704 T + p^{3} T^{2}
83 1−888T+p3T2 1 - 888 T + p^{3} T^{2}
89 1−1020T+p3T2 1 - 1020 T + p^{3} T^{2}
97 1+110T+p3T2 1 + 110 T + p^{3} T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.35799618265716317387053278380, −9.540365643710024220110592462692, −9.024165108235927590371941439577, −7.903894330848768917816924189738, −6.37086881826297221728138093553, −6.04387648420180379982702614490, −4.73841640533161114360269252364, −4.01776500521957705696944555852, −2.20404805590089038910357439652, −0.77878815646777223540205976655, 0.77878815646777223540205976655, 2.20404805590089038910357439652, 4.01776500521957705696944555852, 4.73841640533161114360269252364, 6.04387648420180379982702614490, 6.37086881826297221728138093553, 7.903894330848768917816924189738, 9.024165108235927590371941439577, 9.540365643710024220110592462692, 10.35799618265716317387053278380

Graph of the ZZ-function along the critical line