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Properties

Label 2-13-13.8-c8-0-5
Degree $2$
Conductor $13$
Sign $0.316 + 0.948i$
Analytic cond. $5.29592$
Root an. cond. $2.30128$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.32 + 3.32i)2-s − 9.51·3-s − 233. i·4-s + (−136. − 136. i)5-s + (−31.6 − 31.6i)6-s + (2.56e3 − 2.56e3i)7-s + (1.62e3 − 1.62e3i)8-s − 6.47e3·9-s − 903. i·10-s + (1.62e4 − 1.62e4i)11-s + 2.22e3i·12-s + (−2.33e4 + 1.64e4i)13-s + 1.70e4·14-s + (1.29e3 + 1.29e3i)15-s − 4.90e4·16-s + 1.17e5i·17-s + ⋯
L(s)  = 1  + (0.207 + 0.207i)2-s − 0.117·3-s − 0.913i·4-s + (−0.217 − 0.217i)5-s + (−0.0243 − 0.0243i)6-s + (1.06 − 1.06i)7-s + (0.397 − 0.397i)8-s − 0.986·9-s − 0.0903i·10-s + (1.10 − 1.10i)11-s + 0.107i·12-s + (−0.816 + 0.577i)13-s + 0.443·14-s + (0.0255 + 0.0255i)15-s − 0.748·16-s + 1.41i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.316 + 0.948i$
Analytic conductor: \(5.29592\)
Root analytic conductor: \(2.30128\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :4),\ 0.316 + 0.948i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.30208 - 0.938596i\)
\(L(\frac12)\) \(\approx\) \(1.30208 - 0.938596i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (2.33e4 - 1.64e4i)T \)
good2 \( 1 + (-3.32 - 3.32i)T + 256iT^{2} \)
3 \( 1 + 9.51T + 6.56e3T^{2} \)
5 \( 1 + (136. + 136. i)T + 3.90e5iT^{2} \)
7 \( 1 + (-2.56e3 + 2.56e3i)T - 5.76e6iT^{2} \)
11 \( 1 + (-1.62e4 + 1.62e4i)T - 2.14e8iT^{2} \)
17 \( 1 - 1.17e5iT - 6.97e9T^{2} \)
19 \( 1 + (-5.62e4 - 5.62e4i)T + 1.69e10iT^{2} \)
23 \( 1 + 1.72e5iT - 7.83e10T^{2} \)
29 \( 1 - 1.11e6T + 5.00e11T^{2} \)
31 \( 1 + (-6.24e5 - 6.24e5i)T + 8.52e11iT^{2} \)
37 \( 1 + (1.24e5 - 1.24e5i)T - 3.51e12iT^{2} \)
41 \( 1 + (-1.51e6 - 1.51e6i)T + 7.98e12iT^{2} \)
43 \( 1 - 2.86e6iT - 1.16e13T^{2} \)
47 \( 1 + (-1.31e6 + 1.31e6i)T - 2.38e13iT^{2} \)
53 \( 1 - 1.35e6T + 6.22e13T^{2} \)
59 \( 1 + (-7.29e6 + 7.29e6i)T - 1.46e14iT^{2} \)
61 \( 1 + 1.76e7T + 1.91e14T^{2} \)
67 \( 1 + (-1.13e7 - 1.13e7i)T + 4.06e14iT^{2} \)
71 \( 1 + (1.05e7 + 1.05e7i)T + 6.45e14iT^{2} \)
73 \( 1 + (3.05e7 - 3.05e7i)T - 8.06e14iT^{2} \)
79 \( 1 - 1.90e7T + 1.51e15T^{2} \)
83 \( 1 + (3.52e7 + 3.52e7i)T + 2.25e15iT^{2} \)
89 \( 1 + (-1.60e7 + 1.60e7i)T - 3.93e15iT^{2} \)
97 \( 1 + (-1.08e8 - 1.08e8i)T + 7.83e15iT^{2} \)
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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

−17.47401511000823118265373534639, −16.46306887139573652673939089989, −14.42724827191024530204094232399, −14.13380795999498053601876198354, −11.69802138357199615233309085351, −10.47747125890875335226450552466, −8.425070692550931990541487584366, −6.31546774450382848723775116951, −4.48107870880940017683568994208, −1.04085179613061025179548627336, 2.62430503859719021444994762903, 4.94415383080039242739766285893, 7.48483751977542516847719725836, 9.036853202225239261447642258101, 11.61394289832059655583182624997, 12.04715811009108480580517309172, 14.11445016158035604566471487826, 15.35315131568900441813233249841, 17.24571783784640137116359679890, 17.87986182320462749711669273024

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