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Properties

Label 2-507-39.11-c1-0-12
Degree $2$
Conductor $507$
Sign $0.598 - 0.801i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 − 0.258i)2-s + (1.72 + 0.158i)3-s + (−0.866 − 0.5i)4-s + (−1.41 + 1.41i)5-s + (−1.62 − 0.599i)6-s + (−0.366 − 1.36i)7-s + (2.12 + 2.12i)8-s + (2.94 + 0.548i)9-s + (1.73 − 1.00i)10-s + (−1.03 + 3.86i)11-s + (−1.41 − i)12-s + 1.41i·14-s + (−2.66 + 2.21i)15-s + (−0.500 − 0.866i)16-s + (−2.70 − 1.29i)18-s + (1.36 − 0.366i)19-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.995 + 0.0917i)3-s + (−0.433 − 0.250i)4-s + (−0.632 + 0.632i)5-s + (−0.663 − 0.244i)6-s + (−0.138 − 0.516i)7-s + (0.749 + 0.749i)8-s + (0.983 + 0.182i)9-s + (0.547 − 0.316i)10-s + (−0.312 + 1.16i)11-s + (−0.408 − 0.288i)12-s + 0.377i·14-s + (−0.687 + 0.571i)15-s + (−0.125 − 0.216i)16-s + (−0.638 − 0.304i)18-s + (0.313 − 0.0839i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.598 - 0.801i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.598 - 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.914275 + 0.458413i\)
\(L(\frac12)\) \(\approx\) \(0.914275 + 0.458413i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 0.158i)T \)
13 \( 1 \)
good2 \( 1 + (0.965 + 0.258i)T + (1.73 + i)T^{2} \)
5 \( 1 + (1.41 - 1.41i)T - 5iT^{2} \)
7 \( 1 + (0.366 + 1.36i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.03 - 3.86i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.36 + 0.366i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.24 - 7.34i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.44 - 1.41i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5 - 5i)T + 31iT^{2} \)
37 \( 1 + (-1.36 - 0.366i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.93 - 0.517i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.19 + 3i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.82 + 2.82i)T + 47iT^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 + (-3.86 + 1.03i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.83 + 6.83i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.03 - 3.86i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1 - i)T - 73iT^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + (5.65 - 5.65i)T - 83iT^{2} \)
89 \( 1 + (-3.62 + 13.5i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-9.56 + 2.56i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.68678228414162957041529915484, −10.01197646384619896743728400287, −9.367497509132491453787144809271, −8.446375288687084016064337788268, −7.42529263281387613800383764594, −7.14492750087859401231261337430, −5.14995310440137821461657249426, −4.14065535348421950384462989359, −3.05967910645698190686633613198, −1.55309012757229793953182551329, 0.76090323540885827249053761075, 2.76264011078639729877123693321, 3.90225091853828725743217693368, 4.85745920061309050091066303382, 6.42234536237291531815323183526, 7.66759754694493968703982794575, 8.300285778646559152128166521504, 8.758192475142564603866699747156, 9.527047775912563468852569426431, 10.49231059786163226949129978464

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