↑

Properties

Label 2-13-13.5-c6-0-5
Degree $2$
Conductor $13$
Sign $-0.451 + 0.892i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (9.73 − 9.73i)2-s − 41.3·3-s − 125. i·4-s + (110. − 110. i)5-s + (−402. + 402. i)6-s + (271. + 271. i)7-s + (−597. − 597. i)8-s + 977.·9-s − 2.15e3i·10-s + (523. + 523. i)11-s + 5.17e3i·12-s + (−1.58e3 − 1.51e3i)13-s + 5.28e3·14-s + (−4.58e3 + 4.58e3i)15-s − 3.59e3·16-s + 1.83e3i·17-s + ⋯
L(s)  = 1  + (1.21 − 1.21i)2-s − 1.53·3-s − 1.95i·4-s + (0.887 − 0.887i)5-s + (−1.86 + 1.86i)6-s + (0.791 + 0.791i)7-s + (−1.16 − 1.16i)8-s + 1.34·9-s − 2.15i·10-s + (0.393 + 0.393i)11-s + 2.99i·12-s + (−0.723 − 0.690i)13-s + 1.92·14-s + (−1.35 + 1.35i)15-s − 0.878·16-s + 0.373i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-0.451 + 0.892i$
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: \((2,\ 13,\ (\ :3),\ -0.451 + 0.892i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.964457 - 1.56901i\)
\(L(\frac12)\) \(\approx\) \(0.964457 - 1.56901i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (1.58e3 + 1.51e3i)T \)
good2 \( 1 + (-9.73 + 9.73i)T - 64iT^{2} \)
3 \( 1 + 41.3T + 729T^{2} \)
5 \( 1 + (-110. + 110. i)T - 1.56e4iT^{2} \)
7 \( 1 + (-271. - 271. i)T + 1.17e5iT^{2} \)
11 \( 1 + (-523. - 523. i)T + 1.77e6iT^{2} \)
17 \( 1 - 1.83e3iT - 2.41e7T^{2} \)
19 \( 1 + (-1.99e3 + 1.99e3i)T - 4.70e7iT^{2} \)
23 \( 1 - 2.01e4iT - 1.48e8T^{2} \)
29 \( 1 + 1.29e4T + 5.94e8T^{2} \)
31 \( 1 + (-2.26e3 + 2.26e3i)T - 8.87e8iT^{2} \)
37 \( 1 + (-954. - 954. i)T + 2.56e9iT^{2} \)
41 \( 1 + (3.81e4 - 3.81e4i)T - 4.75e9iT^{2} \)
43 \( 1 - 5.75e4iT - 6.32e9T^{2} \)
47 \( 1 + (1.07e5 + 1.07e5i)T + 1.07e10iT^{2} \)
53 \( 1 - 2.23e5T + 2.21e10T^{2} \)
59 \( 1 + (1.79e5 + 1.79e5i)T + 4.21e10iT^{2} \)
61 \( 1 + 1.55e5T + 5.15e10T^{2} \)
67 \( 1 + (1.83e5 - 1.83e5i)T - 9.04e10iT^{2} \)
71 \( 1 + (1.83e5 - 1.83e5i)T - 1.28e11iT^{2} \)
73 \( 1 + (-1.29e5 - 1.29e5i)T + 1.51e11iT^{2} \)
79 \( 1 - 5.99e5T + 2.43e11T^{2} \)
83 \( 1 + (4.46e5 - 4.46e5i)T - 3.26e11iT^{2} \)
89 \( 1 + (2.08e5 + 2.08e5i)T + 4.96e11iT^{2} \)
97 \( 1 + (-4.77e5 + 4.77e5i)T - 8.32e11iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.00334761830344550927820737945, −17.06030188411160160905395489448, −15.04638575187399298619296596474, −13.25515373529562311138666318674, −12.20861650305198859055933641621, −11.38069749166959453690192147387, −9.830468302415510691239302237314, −5.63582779683439791703237382172, −4.98134388997065816822511361284, −1.54671117598442303582915104988, 4.68370083190392673519968836507, 6.10814302173413882934807115089, 7.11115595625981837329062561836, 10.61094736475461280304780239771, 12.05136961844965554172552668717, 13.80237351278890581531272612957, 14.63676054650395091409276896236, 16.51484573099684440298174114309, 17.12089670538376009096516515822, 18.22713874694166813085724340859

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