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Properties

Label 2-507-13.12-c1-0-4
Degree $2$
Conductor $507$
Sign $-0.832 - 0.554i$
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  + i·2-s − 3-s + 4-s + 2i·5-s − i·6-s + 4i·7-s + 3i·8-s + 9-s − 2·10-s − 4i·11-s − 12-s − 4·14-s − 2i·15-s − 16-s − 2·17-s + i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s + 0.5·4-s + 0.894i·5-s − 0.408i·6-s + 1.51i·7-s + 1.06i·8-s + 0.333·9-s − 0.632·10-s − 1.20i·11-s − 0.288·12-s − 1.06·14-s − 0.516i·15-s − 0.250·16-s − 0.485·17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.832 - 0.554i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.832 - 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.367553 + 1.21394i\)
\(L(\frac12)\) \(\approx\) \(0.367553 + 1.21394i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
13 \( 1 \)
good2 \( 1 - iT - 2T^{2} \)
5 \( 1 - 2iT - 5T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 12iT - 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 2iT - 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−11.15988164067414791761595922941, −10.79091640292724697445636220594, −9.328119828160634463505411049130, −8.475039207531789124869526032260, −7.47885806748749413135659960307, −6.45235019367160744624712390795, −5.95006734907181090512861581927, −5.16203985282114134813121088839, −3.25291631260208456149630665964, −2.20603626271941433523197019764, 0.803801514756463595774766698493, 2.03105277345741485102557666130, 3.89621577422803255855213848135, 4.49955788025706329930696655792, 5.83579496331200400024470915421, 7.20213804215951493017369447649, 7.39656934678084437616839024501, 9.094854179047065514979181005699, 9.949140370334661451655029102257, 10.66282457223276246036277091316

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