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Properties

Label 2-5-5.2-c26-0-5
Degree $2$
Conductor $5$
Sign $-0.935 - 0.354i$
Analytic cond. $21.4146$
Root an. cond. $4.62759$
Motivic weight $26$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.25e3 + 6.25e3i)2-s + (−7.48e5 + 7.48e5i)3-s + 1.11e7i·4-s + (7.43e8 + 9.67e8i)5-s − 9.36e9·6-s + (5.67e10 + 5.67e10i)7-s + (3.50e11 − 3.50e11i)8-s + 1.42e12i·9-s + (−1.40e12 + 1.07e13i)10-s − 7.24e12·11-s + (−8.32e12 − 8.32e12i)12-s + (−3.18e14 + 3.18e14i)13-s + 7.09e14i·14-s + (−1.28e15 − 1.67e14i)15-s + 5.12e15·16-s + (−2.33e15 − 2.33e15i)17-s + ⋯
L(s)  = 1  + (0.763 + 0.763i)2-s + (−0.469 + 0.469i)3-s + 0.165i·4-s + (0.609 + 0.792i)5-s − 0.717·6-s + (0.585 + 0.585i)7-s + (0.636 − 0.636i)8-s + 0.558i·9-s + (−0.140 + 1.07i)10-s − 0.209·11-s + (−0.0778 − 0.0778i)12-s + (−1.05 + 1.05i)13-s + 0.893i·14-s + (−0.658 − 0.0861i)15-s + 1.13·16-s + (−0.235 − 0.235i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.935 - 0.354i$
Analytic conductor: \(21.4146\)
Root analytic conductor: \(4.62759\)
Motivic weight: \(26\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :13),\ -0.935 - 0.354i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(0.443971 + 2.42640i\)
\(L(\frac12)\) \(\approx\) \(0.443971 + 2.42640i\)
\(L(14)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-7.43e8 - 9.67e8i)T \)
good2 \( 1 + (-6.25e3 - 6.25e3i)T + 6.71e7iT^{2} \)
3 \( 1 + (7.48e5 - 7.48e5i)T - 2.54e12iT^{2} \)
7 \( 1 + (-5.67e10 - 5.67e10i)T + 9.38e21iT^{2} \)
11 \( 1 + 7.24e12T + 1.19e27T^{2} \)
13 \( 1 + (3.18e14 - 3.18e14i)T - 9.17e28iT^{2} \)
17 \( 1 + (2.33e15 + 2.33e15i)T + 9.81e31iT^{2} \)
19 \( 1 + 5.21e16iT - 1.76e33T^{2} \)
23 \( 1 + (5.96e17 - 5.96e17i)T - 2.54e35iT^{2} \)
29 \( 1 - 5.57e18iT - 1.05e38T^{2} \)
31 \( 1 - 2.76e18T + 5.96e38T^{2} \)
37 \( 1 + (-1.45e20 - 1.45e20i)T + 5.93e40iT^{2} \)
41 \( 1 - 7.08e20T + 8.55e41T^{2} \)
43 \( 1 + (1.15e20 - 1.15e20i)T - 2.95e42iT^{2} \)
47 \( 1 + (-4.77e21 - 4.77e21i)T + 2.98e43iT^{2} \)
53 \( 1 + (-8.79e21 + 8.79e21i)T - 6.77e44iT^{2} \)
59 \( 1 + 1.00e22iT - 1.10e46T^{2} \)
61 \( 1 - 2.85e23T + 2.62e46T^{2} \)
67 \( 1 + (-2.54e23 - 2.54e23i)T + 3.00e47iT^{2} \)
71 \( 1 + 3.34e23T + 1.35e48T^{2} \)
73 \( 1 + (2.01e24 - 2.01e24i)T - 2.79e48iT^{2} \)
79 \( 1 + 6.89e24iT - 2.17e49T^{2} \)
83 \( 1 + (-8.09e24 + 8.09e24i)T - 7.87e49iT^{2} \)
89 \( 1 - 3.36e25iT - 4.83e50T^{2} \)
97 \( 1 + (5.11e25 + 5.11e25i)T + 4.52e51iT^{2} \)
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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

−17.55605167466128754304071284153, −15.99209304052950139687143865578, −14.73787151809794181212668061161, −13.63153994696764415457843347497, −11.35264381678977421402516764991, −9.816483974723436513614640245030, −7.19814082546402991480896882852, −5.68031436848002206915029010654, −4.62120255753851116116132146750, −2.16248934406428745209752391518, 0.70762486584033717759095055140, 2.15650618752346600297106988261, 4.20486069253409071792984331516, 5.65875920456801671770370640371, 7.931961486994472610538631294507, 10.29936215057737903573148260854, 12.08146094980956700788307239322, 12.86331062070558215213339777833, 14.34341764330916050376815613560, 16.87514712995222929683614780929

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