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Properties

Label 2-13-13.5-c4-0-1
Degree $2$
Conductor $13$
Sign $0.867 - 0.497i$
Analytic cond. $1.34380$
Root an. cond. $1.15922$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 + 1.30i)2-s + 9.97·3-s + 12.5i·4-s + (9.28 − 9.28i)5-s + (−13.0 + 13.0i)6-s + (−46.1 − 46.1i)7-s + (−37.3 − 37.3i)8-s + 18.4·9-s + 24.2i·10-s + (90.5 + 90.5i)11-s + 125. i·12-s + (−37.9 − 164. i)13-s + 120.·14-s + (92.5 − 92.5i)15-s − 103.·16-s + 298. i·17-s + ⋯
L(s)  = 1  + (−0.326 + 0.326i)2-s + 1.10·3-s + 0.786i·4-s + (0.371 − 0.371i)5-s + (−0.361 + 0.361i)6-s + (−0.942 − 0.942i)7-s + (−0.583 − 0.583i)8-s + 0.228·9-s + 0.242i·10-s + (0.748 + 0.748i)11-s + 0.871i·12-s + (−0.224 − 0.974i)13-s + 0.615·14-s + (0.411 − 0.411i)15-s − 0.405·16-s + 1.03i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.867 - 0.497i$
Analytic conductor: \(1.34380\)
Root analytic conductor: \(1.15922\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :2),\ 0.867 - 0.497i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.17716 + 0.313446i\)
\(L(\frac12)\) \(\approx\) \(1.17716 + 0.313446i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (37.9 + 164. i)T \)
good2 \( 1 + (1.30 - 1.30i)T - 16iT^{2} \)
3 \( 1 - 9.97T + 81T^{2} \)
5 \( 1 + (-9.28 + 9.28i)T - 625iT^{2} \)
7 \( 1 + (46.1 + 46.1i)T + 2.40e3iT^{2} \)
11 \( 1 + (-90.5 - 90.5i)T + 1.46e4iT^{2} \)
17 \( 1 - 298. iT - 8.35e4T^{2} \)
19 \( 1 + (-145. + 145. i)T - 1.30e5iT^{2} \)
23 \( 1 - 72.2iT - 2.79e5T^{2} \)
29 \( 1 - 1.45e3T + 7.07e5T^{2} \)
31 \( 1 + (974. - 974. i)T - 9.23e5iT^{2} \)
37 \( 1 + (-345. - 345. i)T + 1.87e6iT^{2} \)
41 \( 1 + (546. - 546. i)T - 2.82e6iT^{2} \)
43 \( 1 + 2.28e3iT - 3.41e6T^{2} \)
47 \( 1 + (1.93e3 + 1.93e3i)T + 4.87e6iT^{2} \)
53 \( 1 - 4.35e3T + 7.89e6T^{2} \)
59 \( 1 + (1.54e3 + 1.54e3i)T + 1.21e7iT^{2} \)
61 \( 1 + 897.T + 1.38e7T^{2} \)
67 \( 1 + (-2.23e3 + 2.23e3i)T - 2.01e7iT^{2} \)
71 \( 1 + (1.21e3 - 1.21e3i)T - 2.54e7iT^{2} \)
73 \( 1 + (-1.61e3 - 1.61e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 1.23e3T + 3.89e7T^{2} \)
83 \( 1 + (2.62e3 - 2.62e3i)T - 4.74e7iT^{2} \)
89 \( 1 + (-2.84e3 - 2.84e3i)T + 6.27e7iT^{2} \)
97 \( 1 + (4.18e3 - 4.18e3i)T - 8.85e7iT^{2} \)
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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

−19.66013979090416738051426820026, −17.64729382959346540226387291950, −16.73476232734005123949949572611, −15.22057767828188439665054171266, −13.61299692098360491467554207543, −12.57659551258029871470185406246, −9.867523882550041331979556102007, −8.581337885125100114937737405767, −7.06802492569208739811394267935, −3.47382028914798590928897245272, 2.65095457707160914672818633496, 6.21978837160010328861730921704, 8.912159148923288794444411086608, 9.697541005144794111392895199832, 11.69909679944784611535576291758, 13.86210344084212507157837099505, 14.62110535218605710311348859223, 16.13705164048863454477709532547, 18.33444149792405775524873612743, 19.19039753202410879103116485267

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