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Properties

Label 2-507-39.5-c1-0-30
Degree $2$
Conductor $507$
Sign $-0.944 - 0.328i$
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  + (−1.38 − 1.38i)2-s + (0.526 + 1.65i)3-s + 1.83i·4-s + (1.04 + 1.04i)5-s + (1.55 − 3.01i)6-s + (−3.17 − 3.17i)7-s + (−0.233 + 0.233i)8-s + (−2.44 + 1.73i)9-s − 2.89i·10-s + (0.108 − 0.108i)11-s + (−3.02 + 0.964i)12-s + 8.77i·14-s + (−1.17 + 2.27i)15-s + 4.30·16-s − 3.16·17-s + (5.78 + 0.980i)18-s + ⋯
L(s)  = 1  + (−0.978 − 0.978i)2-s + (0.303 + 0.952i)3-s + 0.915i·4-s + (0.468 + 0.468i)5-s + (0.634 − 1.22i)6-s + (−1.19 − 1.19i)7-s + (−0.0825 + 0.0825i)8-s + (−0.815 + 0.579i)9-s − 0.916i·10-s + (0.0326 − 0.0326i)11-s + (−0.872 + 0.278i)12-s + 2.34i·14-s + (−0.303 + 0.588i)15-s + 1.07·16-s − 0.768·17-s + (1.36 + 0.231i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.944 - 0.328i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.944 - 0.328i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00722098 + 0.0427439i\)
\(L(\frac12)\) \(\approx\) \(0.00722098 + 0.0427439i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.526 - 1.65i)T \)
13 \( 1 \)
good2 \( 1 + (1.38 + 1.38i)T + 2iT^{2} \)
5 \( 1 + (-1.04 - 1.04i)T + 5iT^{2} \)
7 \( 1 + (3.17 + 3.17i)T + 7iT^{2} \)
11 \( 1 + (-0.108 + 0.108i)T - 11iT^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
19 \( 1 + (0.846 - 0.846i)T - 19iT^{2} \)
23 \( 1 + 6.70T + 23T^{2} \)
29 \( 1 + 1.98iT - 29T^{2} \)
31 \( 1 + (3.64 - 3.64i)T - 31iT^{2} \)
37 \( 1 + (2.31 + 2.31i)T + 37iT^{2} \)
41 \( 1 + (5.91 + 5.91i)T + 41iT^{2} \)
43 \( 1 + 2.78iT - 43T^{2} \)
47 \( 1 + (4.06 - 4.06i)T - 47iT^{2} \)
53 \( 1 + 0.628iT - 53T^{2} \)
59 \( 1 + (6.20 - 6.20i)T - 59iT^{2} \)
61 \( 1 - 5.83T + 61T^{2} \)
67 \( 1 + (0.475 - 0.475i)T - 67iT^{2} \)
71 \( 1 + (-8.09 - 8.09i)T + 71iT^{2} \)
73 \( 1 + (-3.92 - 3.92i)T + 73iT^{2} \)
79 \( 1 + 5.46T + 79T^{2} \)
83 \( 1 + (6.40 + 6.40i)T + 83iT^{2} \)
89 \( 1 + (-3.75 + 3.75i)T - 89iT^{2} \)
97 \( 1 + (3.16 - 3.16i)T - 97iT^{2} \)
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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

−10.34602856235785971954186393263, −9.819183391502819136174450082265, −9.064108837772882714081451751652, −8.119157292659729468649456848322, −6.85016951485402377253142552473, −5.78680428581055697862876506189, −4.12756462024783441848122085319, −3.29794349060168741864790432095, −2.17636889430612410258970215670, −0.03131365278798125870326264468, 1.94989531585167266542014301231, 3.30263667294781629368571710495, 5.46808945163583249573006567477, 6.27059800229284317101392340849, 6.77318262285283532374590761679, 7.945096157232979709106362249103, 8.690990472244202793574409908995, 9.294723110985465357629459041130, 9.896148248455578173442336034349, 11.52517193324928728466684833092

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