Properties

Label 2-507-39.5-c1-0-36
Degree $2$
Conductor $507$
Sign $-0.944 - 0.327i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.944 - 0.327i$
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: \((2,\ 507,\ (\ :1/2),\ -0.944 - 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.141569 + 0.840813i\)
\(L(\frac12)\) \(\approx\) \(0.141569 + 0.840813i\)
\(L(\frac{3}{2})\) 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

−10.38944185757482115527260778924, −9.320084431193250199589874343840, −8.743935436451837376555831432215, −8.166219684321302425145448990359, −6.88118478051464889277968739741, −6.01277582964385769881732186966, −4.24048787821216242431788172293, −3.19355799776355652363837878802, −1.86409971757795049891123667437, −0.58545186054593009240629220487, 2.43264035447499391738041254986, 3.65424075026996709634827584185, 4.56750075553390732978128345481, 6.30202954267422248234760012087, 7.06184607916575008035406751528, 7.85268065540074910227855693081, 8.902019208091325792050765311449, 9.231274952455898055982742412943, 10.17229851684325048006872814343, 11.19863105912816197112772435643

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