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Properties

Label 2-13-13.5-c10-0-1
Degree $2$
Conductor $13$
Sign $0.0139 - 0.999i$
Analytic cond. $8.25964$
Root an. cond. $2.87395$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (14.4 − 14.4i)2-s + 144.·3-s + 606. i·4-s + (−4.15e3 + 4.15e3i)5-s + (2.08e3 − 2.08e3i)6-s + (−3.75e3 − 3.75e3i)7-s + (2.35e4 + 2.35e4i)8-s − 3.81e4·9-s + 1.20e5i·10-s + (3.31e4 + 3.31e4i)11-s + 8.77e4i·12-s + (3.53e5 + 1.12e5i)13-s − 1.08e5·14-s + (−6.01e5 + 6.01e5i)15-s + 5.93e4·16-s + 8.53e5i·17-s + ⋯
L(s)  = 1  + (0.451 − 0.451i)2-s + 0.595·3-s + 0.592i·4-s + (−1.33 + 1.33i)5-s + (0.268 − 0.268i)6-s + (−0.223 − 0.223i)7-s + (0.718 + 0.718i)8-s − 0.645·9-s + 1.20i·10-s + (0.205 + 0.205i)11-s + 0.352i·12-s + (0.952 + 0.303i)13-s − 0.201·14-s + (−0.791 + 0.791i)15-s + 0.0566·16-s + 0.601i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0139 - 0.999i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.0139 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.0139 - 0.999i$
Analytic conductor: \(8.25964\)
Root analytic conductor: \(2.87395\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :5),\ 0.0139 - 0.999i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.23959 + 1.22238i\)
\(L(\frac12)\) \(\approx\) \(1.23959 + 1.22238i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.53e5 - 1.12e5i)T \)
good2 \( 1 + (-14.4 + 14.4i)T - 1.02e3iT^{2} \)
3 \( 1 - 144.T + 5.90e4T^{2} \)
5 \( 1 + (4.15e3 - 4.15e3i)T - 9.76e6iT^{2} \)
7 \( 1 + (3.75e3 + 3.75e3i)T + 2.82e8iT^{2} \)
11 \( 1 + (-3.31e4 - 3.31e4i)T + 2.59e10iT^{2} \)
17 \( 1 - 8.53e5iT - 2.01e12T^{2} \)
19 \( 1 + (-2.26e6 + 2.26e6i)T - 6.13e12iT^{2} \)
23 \( 1 - 2.27e6iT - 4.14e13T^{2} \)
29 \( 1 - 1.91e7T + 4.20e14T^{2} \)
31 \( 1 + (3.56e7 - 3.56e7i)T - 8.19e14iT^{2} \)
37 \( 1 + (-5.94e7 - 5.94e7i)T + 4.80e15iT^{2} \)
41 \( 1 + (-3.98e7 + 3.98e7i)T - 1.34e16iT^{2} \)
43 \( 1 - 4.71e7iT - 2.16e16T^{2} \)
47 \( 1 + (-5.60e6 - 5.60e6i)T + 5.25e16iT^{2} \)
53 \( 1 - 3.21e8T + 1.74e17T^{2} \)
59 \( 1 + (-9.18e8 - 9.18e8i)T + 5.11e17iT^{2} \)
61 \( 1 + 1.80e7T + 7.13e17T^{2} \)
67 \( 1 + (-3.93e8 + 3.93e8i)T - 1.82e18iT^{2} \)
71 \( 1 + (-5.30e8 + 5.30e8i)T - 3.25e18iT^{2} \)
73 \( 1 + (1.10e8 + 1.10e8i)T + 4.29e18iT^{2} \)
79 \( 1 + 3.77e9T + 9.46e18T^{2} \)
83 \( 1 + (-1.19e9 + 1.19e9i)T - 1.55e19iT^{2} \)
89 \( 1 + (-4.77e9 - 4.77e9i)T + 3.11e19iT^{2} \)
97 \( 1 + (3.96e9 - 3.96e9i)T - 7.37e19iT^{2} \)
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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

−17.97603939373934135594620312664, −16.12861750395378352645111411053, −14.76730218454981999262742748950, −13.62840676045303865842931872118, −11.83357079239417271677342274103, −10.94187540750967407312373965142, −8.411721654345864630121343281659, −7.10145011322953835401467683664, −3.83044031325449728673482873921, −2.92800837738728093709736323407, 0.75274745589479018012112747852, 3.87484094286712663152489495528, 5.56525325348018882061717335932, 7.88261865206780982013420927247, 9.165890382038336301685914742404, 11.48602182307015853796830439194, 13.02795406801703206283735921738, 14.41055371750294404016234356273, 15.72862552986528720155928731284, 16.43135908008217593208590924031

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