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Properties

Label 2-13-13.5-c6-0-4
Degree $2$
Conductor $13$
Sign $0.590 + 0.807i$
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  + (7.57 − 7.57i)2-s + 26.7·3-s − 50.8i·4-s + (−59.0 + 59.0i)5-s + (202. − 202. i)6-s + (−226. − 226. i)7-s + (99.4 + 99.4i)8-s − 11.8·9-s + 894. i·10-s + (1.07e3 + 1.07e3i)11-s − 1.36e3i·12-s + (−2.07e3 + 727. i)13-s − 3.44e3·14-s + (−1.58e3 + 1.58e3i)15-s + 4.76e3·16-s − 1.50e3i·17-s + ⋯
L(s)  = 1  + (0.947 − 0.947i)2-s + 0.991·3-s − 0.794i·4-s + (−0.472 + 0.472i)5-s + (0.939 − 0.939i)6-s + (−0.661 − 0.661i)7-s + (0.194 + 0.194i)8-s − 0.0163·9-s + 0.894i·10-s + (0.810 + 0.810i)11-s − 0.788i·12-s + (−0.943 + 0.330i)13-s − 1.25·14-s + (−0.468 + 0.468i)15-s + 1.16·16-s − 0.306i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 + 0.807i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.590 + 0.807i$
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: \((2,\ 13,\ (\ :3),\ 0.590 + 0.807i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.18653 - 1.10997i\)
\(L(\frac12)\) \(\approx\) \(2.18653 - 1.10997i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (2.07e3 - 727. i)T \)
good2 \( 1 + (-7.57 + 7.57i)T - 64iT^{2} \)
3 \( 1 - 26.7T + 729T^{2} \)
5 \( 1 + (59.0 - 59.0i)T - 1.56e4iT^{2} \)
7 \( 1 + (226. + 226. i)T + 1.17e5iT^{2} \)
11 \( 1 + (-1.07e3 - 1.07e3i)T + 1.77e6iT^{2} \)
17 \( 1 + 1.50e3iT - 2.41e7T^{2} \)
19 \( 1 + (-4.28e3 + 4.28e3i)T - 4.70e7iT^{2} \)
23 \( 1 + 1.92e4iT - 1.48e8T^{2} \)
29 \( 1 + 3.46e4T + 5.94e8T^{2} \)
31 \( 1 + (-1.44e4 + 1.44e4i)T - 8.87e8iT^{2} \)
37 \( 1 + (-1.79e4 - 1.79e4i)T + 2.56e9iT^{2} \)
41 \( 1 + (-7.23e4 + 7.23e4i)T - 4.75e9iT^{2} \)
43 \( 1 - 1.22e5iT - 6.32e9T^{2} \)
47 \( 1 + (-7.86e4 - 7.86e4i)T + 1.07e10iT^{2} \)
53 \( 1 + 3.12e4T + 2.21e10T^{2} \)
59 \( 1 + (2.62e4 + 2.62e4i)T + 4.21e10iT^{2} \)
61 \( 1 + 2.95e3T + 5.15e10T^{2} \)
67 \( 1 + (2.29e5 - 2.29e5i)T - 9.04e10iT^{2} \)
71 \( 1 + (5.88e4 - 5.88e4i)T - 1.28e11iT^{2} \)
73 \( 1 + (-4.97e5 - 4.97e5i)T + 1.51e11iT^{2} \)
79 \( 1 - 4.80e4T + 2.43e11T^{2} \)
83 \( 1 + (4.11e5 - 4.11e5i)T - 3.26e11iT^{2} \)
89 \( 1 + (2.44e5 + 2.44e5i)T + 4.96e11iT^{2} \)
97 \( 1 + (-4.17e5 + 4.17e5i)T - 8.32e11iT^{2} \)
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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

−19.18280023147298430494479705383, −16.94984652935190666484242201397, −14.88859006010300034334601832568, −14.07713385765118871880457404850, −12.74843795346942964848017950407, −11.34920248099804793315911916025, −9.595232165579712465739490268684, −7.34990252163268599240071750026, −4.14059153918663501922715523942, −2.69617062528233393597593469683, 3.56304407949843614556580986971, 5.74668540480848899089570389690, 7.74498651162475039748508231470, 9.301583789119555957804511153892, 12.15564475576696927056078173645, 13.55833049039372531460162728658, 14.65049217552330244554875715996, 15.65103473008008315892583308653, 16.82064130656042076258867815191, 19.18475650832167749136754972024

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