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Properties

Label 2-13-13.8-c8-0-4
Degree $2$
Conductor $13$
Sign $-0.0953 - 0.995i$
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  + (21.3 + 21.3i)2-s + 58.3·3-s + 651. i·4-s + (−310. − 310. i)5-s + (1.24e3 + 1.24e3i)6-s + (2.55e3 − 2.55e3i)7-s + (−8.42e3 + 8.42e3i)8-s − 3.16e3·9-s − 1.32e4i·10-s + (−5.77e3 + 5.77e3i)11-s + 3.79e4i·12-s + (2.64e4 − 1.08e4i)13-s + 1.08e5·14-s + (−1.80e4 − 1.80e4i)15-s − 1.92e5·16-s − 9.12e4i·17-s + ⋯
L(s)  = 1  + (1.33 + 1.33i)2-s + 0.719·3-s + 2.54i·4-s + (−0.496 − 0.496i)5-s + (0.958 + 0.958i)6-s + (1.06 − 1.06i)7-s + (−2.05 + 2.05i)8-s − 0.481·9-s − 1.32i·10-s + (−0.394 + 0.394i)11-s + 1.83i·12-s + (0.925 − 0.379i)13-s + 2.82·14-s + (−0.357 − 0.357i)15-s − 2.93·16-s − 1.09i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0953 - 0.995i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0953 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-0.0953 - 0.995i$
Analytic conductor: \(5.29592\)
Root analytic conductor: \(2.30128\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :4),\ -0.0953 - 0.995i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.26495 + 2.49224i\)
\(L(\frac12)\) \(\approx\) \(2.26495 + 2.49224i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-2.64e4 + 1.08e4i)T \)
good2 \( 1 + (-21.3 - 21.3i)T + 256iT^{2} \)
3 \( 1 - 58.3T + 6.56e3T^{2} \)
5 \( 1 + (310. + 310. i)T + 3.90e5iT^{2} \)
7 \( 1 + (-2.55e3 + 2.55e3i)T - 5.76e6iT^{2} \)
11 \( 1 + (5.77e3 - 5.77e3i)T - 2.14e8iT^{2} \)
17 \( 1 + 9.12e4iT - 6.97e9T^{2} \)
19 \( 1 + (-7.13e4 - 7.13e4i)T + 1.69e10iT^{2} \)
23 \( 1 - 2.15e5iT - 7.83e10T^{2} \)
29 \( 1 + 9.16e5T + 5.00e11T^{2} \)
31 \( 1 + (-2.00e5 - 2.00e5i)T + 8.52e11iT^{2} \)
37 \( 1 + (1.78e6 - 1.78e6i)T - 3.51e12iT^{2} \)
41 \( 1 + (2.16e6 + 2.16e6i)T + 7.98e12iT^{2} \)
43 \( 1 - 2.68e6iT - 1.16e13T^{2} \)
47 \( 1 + (1.53e6 - 1.53e6i)T - 2.38e13iT^{2} \)
53 \( 1 - 1.09e7T + 6.22e13T^{2} \)
59 \( 1 + (-4.08e6 + 4.08e6i)T - 1.46e14iT^{2} \)
61 \( 1 - 1.40e7T + 1.91e14T^{2} \)
67 \( 1 + (6.53e5 + 6.53e5i)T + 4.06e14iT^{2} \)
71 \( 1 + (6.95e6 + 6.95e6i)T + 6.45e14iT^{2} \)
73 \( 1 + (7.01e6 - 7.01e6i)T - 8.06e14iT^{2} \)
79 \( 1 - 4.87e7T + 1.51e15T^{2} \)
83 \( 1 + (-1.56e7 - 1.56e7i)T + 2.25e15iT^{2} \)
89 \( 1 + (2.63e6 - 2.63e6i)T - 3.93e15iT^{2} \)
97 \( 1 + (-2.21e7 - 2.21e7i)T + 7.83e15iT^{2} \)
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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

−17.76459409998675976373843919779, −16.51356261064430805964068751457, −15.33869109449716321717299275103, −14.15150343245468405697567309203, −13.41571072968467879691907836496, −11.66219684669114701353606748252, −8.336918668981246442114220072839, −7.46600750654230892019742917883, −5.16406961033509648662910748749, −3.67281150933524152840610968049, 2.10158591761409269204667990133, 3.56982792856184175731776008990, 5.55620453396861005615615353051, 8.695811800713136639413009192469, 10.93709475505070901762564202717, 11.74106791456610631145866771881, 13.37101046281103636142232568150, 14.59412226808058224059989022390, 15.25641947894382457549289063788, 18.44608244371013558197728155393

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