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Properties

Label 2-507-13.12-c3-0-71
Degree $2$
Conductor $507$
Sign $-0.999 - 0.0304i$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.34i·2-s + 3·3-s + 2.49·4-s − 15.3i·5-s − 7.04i·6-s − 10.1i·7-s − 24.6i·8-s + 9·9-s − 36.1·10-s + 15.0i·11-s + 7.47·12-s − 23.7·14-s − 46.1i·15-s − 37.8·16-s − 90.8·17-s − 21.1i·18-s + ⋯
L(s)  = 1  − 0.829i·2-s + 0.577·3-s + 0.311·4-s − 1.37i·5-s − 0.479i·6-s − 0.547i·7-s − 1.08i·8-s + 0.333·9-s − 1.14·10-s + 0.412i·11-s + 0.179·12-s − 0.453·14-s − 0.795i·15-s − 0.591·16-s − 1.29·17-s − 0.276i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.999 - 0.0304i$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ -0.999 - 0.0304i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.394448957\)
\(L(\frac12)\) \(\approx\) \(2.394448957\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
13 \( 1 \)
good2 \( 1 + 2.34iT - 8T^{2} \)
5 \( 1 + 15.3iT - 125T^{2} \)
7 \( 1 + 10.1iT - 343T^{2} \)
11 \( 1 - 15.0iT - 1.33e3T^{2} \)
17 \( 1 + 90.8T + 4.91e3T^{2} \)
19 \( 1 + 114. iT - 6.85e3T^{2} \)
23 \( 1 + 75.7T + 1.21e4T^{2} \)
29 \( 1 - 214.T + 2.43e4T^{2} \)
31 \( 1 - 284. iT - 2.97e4T^{2} \)
37 \( 1 - 358. iT - 5.06e4T^{2} \)
41 \( 1 + 313. iT - 6.89e4T^{2} \)
43 \( 1 - 296.T + 7.95e4T^{2} \)
47 \( 1 + 316. iT - 1.03e5T^{2} \)
53 \( 1 - 163.T + 1.48e5T^{2} \)
59 \( 1 - 254. iT - 2.05e5T^{2} \)
61 \( 1 + 935.T + 2.26e5T^{2} \)
67 \( 1 - 240. iT - 3.00e5T^{2} \)
71 \( 1 + 947. iT - 3.57e5T^{2} \)
73 \( 1 + 430. iT - 3.89e5T^{2} \)
79 \( 1 + 496.T + 4.93e5T^{2} \)
83 \( 1 + 392. iT - 5.71e5T^{2} \)
89 \( 1 - 979. iT - 7.04e5T^{2} \)
97 \( 1 - 553. iT - 9.12e5T^{2} \)
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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

−10.16977202749978132741821658477, −9.101634294233635605857246318450, −8.618853021959447514557278261650, −7.34265481538289713583307822320, −6.53721181837401168475911812884, −4.83747331131706665565132192049, −4.21059989760897578783106922866, −2.85753515568343167059604329901, −1.71949771461530087667320141363, −0.64194401174575200849422610276, 2.14616693337637971555983709280, 2.84839065352901469629526022005, 4.18895043133011791050541733553, 5.90310315407188036536438470854, 6.30844984314793792008262075826, 7.35596323036126915503187222738, 8.008928807824336448030917062571, 8.931087301953767370050571422708, 10.09048755029796012538414971923, 10.93980406757771899454872003191

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