Properties

Label 2-507-13.12-c3-0-43
Degree $2$
Conductor $507$
Sign $0.832 + 0.554i$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 8·4-s − 12i·5-s − 2i·7-s + 9·9-s + 36i·11-s − 24·12-s + 36i·15-s + 64·16-s + 78·17-s + 74i·19-s − 96i·20-s + 6i·21-s + 96·23-s − 19·25-s + ⋯
L(s)  = 1  − 0.577·3-s + 4-s − 1.07i·5-s − 0.107i·7-s + 0.333·9-s + 0.986i·11-s − 0.577·12-s + 0.619i·15-s + 16-s + 1.11·17-s + 0.893i·19-s − 1.07i·20-s + 0.0623i·21-s + 0.870·23-s − 0.151·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.832 + 0.554i$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ 0.832 + 0.554i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.214337380\)
\(L(\frac12)\) \(\approx\) \(2.214337380\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
13 \( 1 \)
good2 \( 1 - 8T^{2} \)
5 \( 1 + 12iT - 125T^{2} \)
7 \( 1 + 2iT - 343T^{2} \)
11 \( 1 - 36iT - 1.33e3T^{2} \)
17 \( 1 - 78T + 4.91e3T^{2} \)
19 \( 1 - 74iT - 6.85e3T^{2} \)
23 \( 1 - 96T + 1.21e4T^{2} \)
29 \( 1 - 18T + 2.43e4T^{2} \)
31 \( 1 + 214iT - 2.97e4T^{2} \)
37 \( 1 - 286iT - 5.06e4T^{2} \)
41 \( 1 + 384iT - 6.89e4T^{2} \)
43 \( 1 + 524T + 7.95e4T^{2} \)
47 \( 1 + 300iT - 1.03e5T^{2} \)
53 \( 1 - 558T + 1.48e5T^{2} \)
59 \( 1 + 576iT - 2.05e5T^{2} \)
61 \( 1 - 74T + 2.26e5T^{2} \)
67 \( 1 - 38iT - 3.00e5T^{2} \)
71 \( 1 + 456iT - 3.57e5T^{2} \)
73 \( 1 - 682iT - 3.89e5T^{2} \)
79 \( 1 - 704T + 4.93e5T^{2} \)
83 \( 1 + 888iT - 5.71e5T^{2} \)
89 \( 1 - 1.02e3iT - 7.04e5T^{2} \)
97 \( 1 - 110iT - 9.12e5T^{2} \)
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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

−10.31709816777718576249370920241, −9.824834403123852412048788783493, −8.526892998998024443091679029037, −7.58446194570514473587095502389, −6.77493939022990281976704007702, −5.65954133900115288703694487464, −4.92616042861740135160350351223, −3.62383725782463126359397899111, −1.98206246669603307863181702158, −0.922803367563866036483708166567, 1.06986029452229188950017688255, 2.69597295464047037580463171420, 3.42044431367593246000085483793, 5.19105321088377414643194425256, 6.11274010954071403906120831092, 6.85770087983768156585641910833, 7.54196263282179002308020102137, 8.760890457466282369360590478108, 10.10697745563384368033647550658, 10.73553956874182388077062748758

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