Properties

Label 2-13-13.5-c6-0-3
Degree $2$
Conductor $13$
Sign $-0.0343 + 0.999i$
Analytic cond. $2.99070$
Root an. cond. $1.72936$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8.50 + 8.50i)2-s − 17.1·3-s − 80.5i·4-s + (73.6 − 73.6i)5-s + (145. − 145. i)6-s + (−214. − 214. i)7-s + (140. + 140. i)8-s − 436.·9-s + 1.25e3i·10-s + (231. + 231. i)11-s + 1.37e3i·12-s + (−2.07e3 − 708. i)13-s + 3.64e3·14-s + (−1.26e3 + 1.26e3i)15-s + 2.76e3·16-s − 7.52e3i·17-s + ⋯
L(s)  = 1  + (−1.06 + 1.06i)2-s − 0.633·3-s − 1.25i·4-s + (0.589 − 0.589i)5-s + (0.673 − 0.673i)6-s + (−0.624 − 0.624i)7-s + (0.275 + 0.275i)8-s − 0.598·9-s + 1.25i·10-s + (0.174 + 0.174i)11-s + 0.797i·12-s + (−0.946 − 0.322i)13-s + 1.32·14-s + (−0.373 + 0.373i)15-s + 0.673·16-s − 1.53i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0343 + 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0343 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-0.0343 + 0.999i$
Analytic conductor: \(2.99070\)
Root analytic conductor: \(1.72936\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :3),\ -0.0343 + 0.999i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.154823 - 0.160231i\)
\(L(\frac12)\) \(\approx\) \(0.154823 - 0.160231i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (2.07e3 + 708. i)T \)
good2 \( 1 + (8.50 - 8.50i)T - 64iT^{2} \)
3 \( 1 + 17.1T + 729T^{2} \)
5 \( 1 + (-73.6 + 73.6i)T - 1.56e4iT^{2} \)
7 \( 1 + (214. + 214. i)T + 1.17e5iT^{2} \)
11 \( 1 + (-231. - 231. i)T + 1.77e6iT^{2} \)
17 \( 1 + 7.52e3iT - 2.41e7T^{2} \)
19 \( 1 + (8.30e3 - 8.30e3i)T - 4.70e7iT^{2} \)
23 \( 1 + 2.39e3iT - 1.48e8T^{2} \)
29 \( 1 + 923.T + 5.94e8T^{2} \)
31 \( 1 + (1.64e4 - 1.64e4i)T - 8.87e8iT^{2} \)
37 \( 1 + (3.61e4 + 3.61e4i)T + 2.56e9iT^{2} \)
41 \( 1 + (-4.44e4 + 4.44e4i)T - 4.75e9iT^{2} \)
43 \( 1 + 9.79e4iT - 6.32e9T^{2} \)
47 \( 1 + (-1.12e5 - 1.12e5i)T + 1.07e10iT^{2} \)
53 \( 1 + 5.77e4T + 2.21e10T^{2} \)
59 \( 1 + (5.74e4 + 5.74e4i)T + 4.21e10iT^{2} \)
61 \( 1 - 4.13e5T + 5.15e10T^{2} \)
67 \( 1 + (2.65e5 - 2.65e5i)T - 9.04e10iT^{2} \)
71 \( 1 + (2.88e4 - 2.88e4i)T - 1.28e11iT^{2} \)
73 \( 1 + (3.95e5 + 3.95e5i)T + 1.51e11iT^{2} \)
79 \( 1 + 4.19e5T + 2.43e11T^{2} \)
83 \( 1 + (-5.91e4 + 5.91e4i)T - 3.26e11iT^{2} \)
89 \( 1 + (6.10e4 + 6.10e4i)T + 4.96e11iT^{2} \)
97 \( 1 + (2.35e5 - 2.35e5i)T - 8.32e11iT^{2} \)
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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

−17.62505308827414202313287668769, −16.97088676395882710618588950112, −16.15517354092305736856227346135, −14.36468916834217967267954458043, −12.46609348660127274248062268897, −10.26842134399211451059057481783, −9.002032996561932228357676248868, −7.14924713332811047125809759543, −5.61000219935485713789004375493, −0.24417087655973949509290138804, 2.48428543440956923572269466925, 6.18105194041926002579172058369, 8.814405883545936527889885504105, 10.18857586953912422307366210871, 11.32151525854577912889992329696, 12.63961590592940323807431184493, 14.80285168137269386252993258930, 16.96172770116906911843098648013, 17.67279156804013140637870721558, 19.00320391836345243122741647521

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