Properties

Label 2-507-39.8-c1-0-32
Degree $2$
Conductor $507$
Sign $0.932 + 0.359i$
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  + (1.06 − 1.06i)2-s + (1.36 + 1.06i)3-s − 0.267i·4-s + (1.06 − 1.06i)5-s + (2.58 − 0.320i)6-s + (1 − i)7-s + (1.84 + 1.84i)8-s + (0.732 + 2.90i)9-s − 2.26i·10-s + (−2.90 − 2.90i)11-s + (0.285 − 0.366i)12-s − 2.12i·14-s + (2.58 − 0.320i)15-s + 4.46·16-s − 5.03·17-s + (3.87 + 2.31i)18-s + ⋯
L(s)  = 1  + (0.752 − 0.752i)2-s + (0.788 + 0.614i)3-s − 0.133i·4-s + (0.476 − 0.476i)5-s + (1.05 − 0.130i)6-s + (0.377 − 0.377i)7-s + (0.652 + 0.652i)8-s + (0.244 + 0.969i)9-s − 0.717i·10-s + (−0.877 − 0.877i)11-s + (0.0823 − 0.105i)12-s − 0.569i·14-s + (0.668 − 0.0827i)15-s + 1.11·16-s − 1.22·17-s + (0.913 + 0.546i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.932 + 0.359i$
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: \((2,\ 507,\ (\ :1/2),\ 0.932 + 0.359i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.86228 - 0.532898i\)
\(L(\frac12)\) \(\approx\) \(2.86228 - 0.532898i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.36 - 1.06i)T \)
13 \( 1 \)
good2 \( 1 + (-1.06 + 1.06i)T - 2iT^{2} \)
5 \( 1 + (-1.06 + 1.06i)T - 5iT^{2} \)
7 \( 1 + (-1 + i)T - 7iT^{2} \)
11 \( 1 + (2.90 + 2.90i)T + 11iT^{2} \)
17 \( 1 + 5.03T + 17T^{2} \)
19 \( 1 + (-2.73 - 2.73i)T + 19iT^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 7.16iT - 29T^{2} \)
31 \( 1 + (2.46 + 2.46i)T + 31iT^{2} \)
37 \( 1 + (-3.83 + 3.83i)T - 37iT^{2} \)
41 \( 1 + (3.97 - 3.97i)T - 41iT^{2} \)
43 \( 1 + 2.19iT - 43T^{2} \)
47 \( 1 + (4.25 + 4.25i)T + 47iT^{2} \)
53 \( 1 + 0.779iT - 53T^{2} \)
59 \( 1 + (-2.12 - 2.12i)T + 59iT^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 + (4.19 + 4.19i)T + 67iT^{2} \)
71 \( 1 + (2.12 - 2.12i)T - 71iT^{2} \)
73 \( 1 + (0.901 - 0.901i)T - 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (-2.90 + 2.90i)T - 83iT^{2} \)
89 \( 1 + (6.59 + 6.59i)T + 89iT^{2} \)
97 \( 1 + (1.19 + 1.19i)T + 97iT^{2} \)
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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

−10.95198414802991696957101292982, −10.15171292485844053296679093461, −9.154318785867621140225729363824, −8.222459905895879400605221690755, −7.57790574212608012525415617709, −5.73911111070681762882942283032, −4.84587586974155783040704100822, −3.99998641039788071664614668989, −2.95619030771835067367110588427, −1.89297381903310559239221867201, 1.81752162278138020617486767639, 2.94254894935334599259644899458, 4.47870776692308441017448149064, 5.37013179320599808497001464728, 6.56723316570405305843951644703, 7.06067127043950770358897344460, 7.998350963894187265899303185961, 9.071323402122738379241839899382, 9.992742874938662075944084549952, 10.89651616997035956432906061162

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