Properties

Label 2-507-1.1-c3-0-17
Degree $2$
Conductor $507$
Sign $1$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.425778413\)
\(L(\frac12)\) \(\approx\) \(1.425778413\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

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

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 $Z$-function along the critical line