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Properties

Label 2-13-13.5-c8-0-7
Degree $2$
Conductor $13$
Sign $0.155 + 0.987i$
Analytic cond. $5.29592$
Root an. cond. $2.30128$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (13.2 − 13.2i)2-s + 7.30·3-s − 94.8i·4-s + (788. − 788. i)5-s + (96.6 − 96.6i)6-s + (−1.96e3 − 1.96e3i)7-s + (2.13e3 + 2.13e3i)8-s − 6.50e3·9-s − 2.08e4i·10-s + (1.04e4 + 1.04e4i)11-s − 692. i·12-s + (2.82e4 + 3.93e3i)13-s − 5.20e4·14-s + (5.75e3 − 5.75e3i)15-s + 8.08e4·16-s + 6.63e4i·17-s + ⋯
L(s)  = 1  + (0.827 − 0.827i)2-s + 0.0901·3-s − 0.370i·4-s + (1.26 − 1.26i)5-s + (0.0746 − 0.0746i)6-s + (−0.818 − 0.818i)7-s + (0.521 + 0.521i)8-s − 0.991·9-s − 2.08i·10-s + (0.710 + 0.710i)11-s − 0.0333i·12-s + (0.990 + 0.137i)13-s − 1.35·14-s + (0.113 − 0.113i)15-s + 1.23·16-s + 0.794i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.155 + 0.987i$
Analytic conductor: \(5.29592\)
Root analytic conductor: \(2.30128\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :4),\ 0.155 + 0.987i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.98471 - 1.69736i\)
\(L(\frac12)\) \(\approx\) \(1.98471 - 1.69736i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-2.82e4 - 3.93e3i)T \)
good2 \( 1 + (-13.2 + 13.2i)T - 256iT^{2} \)
3 \( 1 - 7.30T + 6.56e3T^{2} \)
5 \( 1 + (-788. + 788. i)T - 3.90e5iT^{2} \)
7 \( 1 + (1.96e3 + 1.96e3i)T + 5.76e6iT^{2} \)
11 \( 1 + (-1.04e4 - 1.04e4i)T + 2.14e8iT^{2} \)
17 \( 1 - 6.63e4iT - 6.97e9T^{2} \)
19 \( 1 + (1.28e5 - 1.28e5i)T - 1.69e10iT^{2} \)
23 \( 1 - 1.16e5iT - 7.83e10T^{2} \)
29 \( 1 + 3.37e4T + 5.00e11T^{2} \)
31 \( 1 + (4.24e5 - 4.24e5i)T - 8.52e11iT^{2} \)
37 \( 1 + (1.08e5 + 1.08e5i)T + 3.51e12iT^{2} \)
41 \( 1 + (-1.33e6 + 1.33e6i)T - 7.98e12iT^{2} \)
43 \( 1 + 5.44e6iT - 1.16e13T^{2} \)
47 \( 1 + (-1.46e6 - 1.46e6i)T + 2.38e13iT^{2} \)
53 \( 1 - 4.34e6T + 6.22e13T^{2} \)
59 \( 1 + (-4.07e4 - 4.07e4i)T + 1.46e14iT^{2} \)
61 \( 1 + 2.38e7T + 1.91e14T^{2} \)
67 \( 1 + (1.68e7 - 1.68e7i)T - 4.06e14iT^{2} \)
71 \( 1 + (-2.75e6 + 2.75e6i)T - 6.45e14iT^{2} \)
73 \( 1 + (-5.38e6 - 5.38e6i)T + 8.06e14iT^{2} \)
79 \( 1 + 2.15e7T + 1.51e15T^{2} \)
83 \( 1 + (-1.59e6 + 1.59e6i)T - 2.25e15iT^{2} \)
89 \( 1 + (7.12e7 + 7.12e7i)T + 3.93e15iT^{2} \)
97 \( 1 + (-5.18e7 + 5.18e7i)T - 7.83e15iT^{2} \)
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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

−17.26525205356898082551166764419, −16.72795750404488912456048663612, −14.17222539706733871331558293241, −13.25898875180477271055817187569, −12.32529641567669745034981142537, −10.41649447530083176127801498752, −8.803926589619278123238919348727, −5.87027108654845085437272363210, −3.96583613947310360630415049569, −1.66800976048982456849415294158, 2.93170633778047652136089575625, 5.90159476172954177477371984639, 6.46502495126161960157512218511, 9.229225543549293771509540789618, 10.98220452401825863639475168148, 13.30004075583864950427896256903, 14.17669303366967253560104828959, 15.18051798379731529869010311392, 16.66691641179915459060719790638, 18.22424465581134263588044201046

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