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Properties

Label 2-2513-2513.377-c0-0-0
Degree $2$
Conductor $2513$
Sign $-0.314 - 0.949i$
Analytic cond. $1.25415$
Root an. cond. $1.11988$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.08 − 1.62i)2-s + (−1.08 − 2.60i)4-s + (−0.965 + 0.260i)7-s + (−3.50 − 0.684i)8-s + (−0.855 + 0.517i)9-s + (−0.0767 − 1.24i)11-s + (−0.623 + 1.85i)14-s + (−2.92 + 2.95i)16-s + (−0.0857 + 1.95i)18-s + (−2.11 − 1.22i)22-s + (−1.92 − 0.482i)23-s + (0.763 + 0.645i)25-s + (1.72 + 2.23i)28-s + (0.0898 − 0.129i)29-s + (0.914 + 4.46i)32-s + ⋯
L(s)  = 1  + (1.08 − 1.62i)2-s + (−1.08 − 2.60i)4-s + (−0.965 + 0.260i)7-s + (−3.50 − 0.684i)8-s + (−0.855 + 0.517i)9-s + (−0.0767 − 1.24i)11-s + (−0.623 + 1.85i)14-s + (−2.92 + 2.95i)16-s + (−0.0857 + 1.95i)18-s + (−2.11 − 1.22i)22-s + (−1.92 − 0.482i)23-s + (0.763 + 0.645i)25-s + (1.72 + 2.23i)28-s + (0.0898 − 0.129i)29-s + (0.914 + 4.46i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2513\)    =    \(7 \cdot 359\)
Sign: $-0.314 - 0.949i$
Analytic conductor: \(1.25415\)
Root analytic conductor: \(1.11988\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2513} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2513,\ (\ :0),\ -0.314 - 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9496985808\)
\(L(\frac12)\) \(\approx\) \(0.9496985808\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.965 - 0.260i)T \)
359 \( 1 + (-0.926 - 0.376i)T \)
good2 \( 1 + (-1.08 + 1.62i)T + (-0.384 - 0.923i)T^{2} \)
3 \( 1 + (0.855 - 0.517i)T^{2} \)
5 \( 1 + (-0.763 - 0.645i)T^{2} \)
11 \( 1 + (0.0767 + 1.24i)T + (-0.992 + 0.122i)T^{2} \)
13 \( 1 + (-0.432 + 0.901i)T^{2} \)
17 \( 1 + (0.625 - 0.780i)T^{2} \)
19 \( 1 + (-0.554 + 0.832i)T^{2} \)
23 \( 1 + (1.92 + 0.482i)T + (0.881 + 0.471i)T^{2} \)
29 \( 1 + (-0.0898 + 0.129i)T + (-0.352 - 0.935i)T^{2} \)
31 \( 1 + (-0.977 - 0.209i)T^{2} \)
37 \( 1 + (-0.993 + 1.54i)T + (-0.416 - 0.908i)T^{2} \)
41 \( 1 + (-0.984 + 0.174i)T^{2} \)
43 \( 1 + (0.168 + 1.47i)T + (-0.974 + 0.226i)T^{2} \)
47 \( 1 + (0.285 - 0.958i)T^{2} \)
53 \( 1 + (1.40 - 0.599i)T + (0.691 - 0.722i)T^{2} \)
59 \( 1 + (-0.997 - 0.0701i)T^{2} \)
61 \( 1 + (-0.400 + 0.916i)T^{2} \)
67 \( 1 + (0.705 + 1.53i)T + (-0.652 + 0.757i)T^{2} \)
71 \( 1 + (-0.811 - 0.877i)T + (-0.0788 + 0.996i)T^{2} \)
73 \( 1 + (-0.997 + 0.0701i)T^{2} \)
79 \( 1 + (-0.451 - 0.130i)T + (0.846 + 0.532i)T^{2} \)
83 \( 1 + (-0.665 + 0.746i)T^{2} \)
89 \( 1 + (-0.969 - 0.243i)T^{2} \)
97 \( 1 + (0.416 - 0.908i)T^{2} \)
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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

−8.956735838962424781166582677210, −8.082466109543743674538277626695, −6.51092133914077735600530983482, −5.81302959999873612160264871598, −5.44437211248341277578502090240, −4.27612433039971465201763873340, −3.47693415255246465048827046736, −2.83610406918124109466610860956, −2.04198731679958271358048635097, −0.40546256697916569301452337261, 2.65753041195538790721810672559, 3.45552272735470846116536999204, 4.30459013102625305559507164568, 4.97314688168240529967571322807, 6.14275082681359151675218343605, 6.26979586978459945762013075939, 7.12350378420902621007658383307, 7.901865682789600749809842927615, 8.481267013464494384625061197237, 9.501640920989402336802208856434

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