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Properties

Label 2-13-13.8-c6-0-4
Degree $2$
Conductor $13$
Sign $-0.358 + 0.933i$
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  + (−6.85 − 6.85i)2-s + 38.7·3-s + 30.0i·4-s + (−127. − 127. i)5-s + (−265. − 265. i)6-s + (215. − 215. i)7-s + (−232. + 232. i)8-s + 772.·9-s + 1.74e3i·10-s + (458. − 458. i)11-s + 1.16e3i·12-s + (2.19e3 + 159. i)13-s − 2.95e3·14-s + (−4.92e3 − 4.92e3i)15-s + 5.11e3·16-s + 8.24e3i·17-s + ⋯
L(s)  = 1  + (−0.857 − 0.857i)2-s + 1.43·3-s + 0.469i·4-s + (−1.01 − 1.01i)5-s + (−1.23 − 1.23i)6-s + (0.629 − 0.629i)7-s + (−0.454 + 0.454i)8-s + 1.05·9-s + 1.74i·10-s + (0.344 − 0.344i)11-s + 0.673i·12-s + (0.997 + 0.0725i)13-s − 1.07·14-s + (−1.46 − 1.46i)15-s + 1.24·16-s + 1.67i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-0.358 + 0.933i$
Analytic conductor: \(2.99070\)
Root analytic conductor: \(1.72936\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :3),\ -0.358 + 0.933i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.691922 - 1.00690i\)
\(L(\frac12)\) \(\approx\) \(0.691922 - 1.00690i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-2.19e3 - 159. i)T \)
good2 \( 1 + (6.85 + 6.85i)T + 64iT^{2} \)
3 \( 1 - 38.7T + 729T^{2} \)
5 \( 1 + (127. + 127. i)T + 1.56e4iT^{2} \)
7 \( 1 + (-215. + 215. i)T - 1.17e5iT^{2} \)
11 \( 1 + (-458. + 458. i)T - 1.77e6iT^{2} \)
17 \( 1 - 8.24e3iT - 2.41e7T^{2} \)
19 \( 1 + (-2.80e3 - 2.80e3i)T + 4.70e7iT^{2} \)
23 \( 1 + 1.33e4iT - 1.48e8T^{2} \)
29 \( 1 - 1.80e4T + 5.94e8T^{2} \)
31 \( 1 + (-1.06e4 - 1.06e4i)T + 8.87e8iT^{2} \)
37 \( 1 + (5.31e4 - 5.31e4i)T - 2.56e9iT^{2} \)
41 \( 1 + (6.04e4 + 6.04e4i)T + 4.75e9iT^{2} \)
43 \( 1 + 6.03e4iT - 6.32e9T^{2} \)
47 \( 1 + (-3.54e4 + 3.54e4i)T - 1.07e10iT^{2} \)
53 \( 1 - 1.28e5T + 2.21e10T^{2} \)
59 \( 1 + (1.75e5 - 1.75e5i)T - 4.21e10iT^{2} \)
61 \( 1 + 1.25e5T + 5.15e10T^{2} \)
67 \( 1 + (-8.31e4 - 8.31e4i)T + 9.04e10iT^{2} \)
71 \( 1 + (-2.92e5 - 2.92e5i)T + 1.28e11iT^{2} \)
73 \( 1 + (-2.21e5 + 2.21e5i)T - 1.51e11iT^{2} \)
79 \( 1 + 3.66e5T + 2.43e11T^{2} \)
83 \( 1 + (-3.15e5 - 3.15e5i)T + 3.26e11iT^{2} \)
89 \( 1 + (6.76e5 - 6.76e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (3.21e5 + 3.21e5i)T + 8.32e11iT^{2} \)
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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

−18.77890163841327046370461800228, −16.99106006152150087146566828813, −15.30378000522790520288916553349, −13.92356252083316214764902134384, −12.16931267043373817838091520568, −10.55664890306608281007017352265, −8.589182945158370955295995054592, −8.330288237370892661886895983281, −3.80372861088556971791478337985, −1.30315968327935357353648199589, 3.23372779221072048128359094616, 7.15492914187692498824061629093, 8.151353982788181559813652754071, 9.329019391547684124206694084672, 11.61184463594745147052382871541, 13.99856589129211940896075311265, 15.21306287521056691718046656623, 15.77737930873844393214474871451, 17.99842710528506174735609737442, 18.75335381368619548502883202908

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