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Properties

Label 2-507-39.8-c1-0-24
Degree $2$
Conductor $507$
Sign $0.0851 - 0.996i$
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 + (−3.01 − 1.55i)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 + (2.27 + 1.17i)15-s + 4.30·16-s + 3.16·17-s + (0.980 − 5.78i)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 + (−1.22 − 0.634i)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 + (0.588 + 0.303i)15-s + 1.07·16-s + 0.768·17-s + (0.231 − 1.36i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0851 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0851 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.0851 - 0.996i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.0851 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.847577 + 0.778221i\)
\(L(\frac12)\) \(\approx\) \(0.847577 + 0.778221i\)
\(L(\frac{3}{2})\) 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

−10.72232080343679874957063320245, −9.953787999332992958238154472047, −9.290314669453835162415777328141, −8.335146426659272364678853587505, −7.82741565863423187637610655677, −6.82361302221520799136465873263, −5.44618680753210645853657680411, −4.71725171084492593490597023501, −3.41549326124479470305191879210, −1.22881284199260281997043943699, 1.26337277911842990257581246599, 2.27678439519794049457926115872, 3.01513538339014992977011384087, 5.21271139384540587069901073068, 6.12079714625217780407684713300, 7.41107662937402266542587798357, 8.353245567620003554557437401827, 8.841972930762767516612765560027, 9.759481349805865459247222722636, 10.80142542281569395109985015005

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