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Properties

Label 2-5-5.3-c26-0-7
Degree $2$
Conductor $5$
Sign $0.800 + 0.599i$
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  + (−3.78e3 + 3.78e3i)2-s + (1.14e6 + 1.14e6i)3-s + 3.85e7i·4-s + (−1.21e9 − 1.09e8i)5-s − 8.63e9·6-s + (1.09e11 − 1.09e11i)7-s + (−3.99e11 − 3.99e11i)8-s + 6.48e10i·9-s + (5.01e12 − 4.18e12i)10-s − 3.07e13·11-s + (−4.39e13 + 4.39e13i)12-s + (−1.16e14 − 1.16e14i)13-s + 8.31e14i·14-s + (−1.26e15 − 1.51e15i)15-s + 4.35e14·16-s + (1.08e16 − 1.08e16i)17-s + ⋯
L(s)  = 1  + (−0.461 + 0.461i)2-s + (0.716 + 0.716i)3-s + 0.573i·4-s + (−0.995 − 0.0895i)5-s − 0.661·6-s + (1.13 − 1.13i)7-s + (−0.726 − 0.726i)8-s + 0.0254i·9-s + (0.501 − 0.418i)10-s − 0.891·11-s + (−0.410 + 0.410i)12-s + (−0.386 − 0.386i)13-s + 1.04i·14-s + (−0.649 − 0.777i)15-s + 0.0966·16-s + (1.09 − 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.800 + 0.599i$
Analytic conductor: \(21.4146\)
Root analytic conductor: \(4.62759\)
Motivic weight: \(26\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :13),\ 0.800 + 0.599i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(1.01596 - 0.338534i\)
\(L(\frac12)\) \(\approx\) \(1.01596 - 0.338534i\)
\(L(14)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.21e9 + 1.09e8i)T \)
good2 \( 1 + (3.78e3 - 3.78e3i)T - 6.71e7iT^{2} \)
3 \( 1 + (-1.14e6 - 1.14e6i)T + 2.54e12iT^{2} \)
7 \( 1 + (-1.09e11 + 1.09e11i)T - 9.38e21iT^{2} \)
11 \( 1 + 3.07e13T + 1.19e27T^{2} \)
13 \( 1 + (1.16e14 + 1.16e14i)T + 9.17e28iT^{2} \)
17 \( 1 + (-1.08e16 + 1.08e16i)T - 9.81e31iT^{2} \)
19 \( 1 - 5.46e16iT - 1.76e33T^{2} \)
23 \( 1 + (4.13e17 + 4.13e17i)T + 2.54e35iT^{2} \)
29 \( 1 + 1.04e19iT - 1.05e38T^{2} \)
31 \( 1 - 1.88e18T + 5.96e38T^{2} \)
37 \( 1 + (1.92e20 - 1.92e20i)T - 5.93e40iT^{2} \)
41 \( 1 - 3.59e20T + 8.55e41T^{2} \)
43 \( 1 + (1.95e20 + 1.95e20i)T + 2.95e42iT^{2} \)
47 \( 1 + (-1.14e21 + 1.14e21i)T - 2.98e43iT^{2} \)
53 \( 1 + (4.01e21 + 4.01e21i)T + 6.77e44iT^{2} \)
59 \( 1 + 6.50e22iT - 1.10e46T^{2} \)
61 \( 1 + 3.16e23T + 2.62e46T^{2} \)
67 \( 1 + (-2.78e23 + 2.78e23i)T - 3.00e47iT^{2} \)
71 \( 1 - 1.63e23T + 1.35e48T^{2} \)
73 \( 1 + (-7.02e22 - 7.02e22i)T + 2.79e48iT^{2} \)
79 \( 1 + 1.67e24iT - 2.17e49T^{2} \)
83 \( 1 + (9.74e24 + 9.74e24i)T + 7.87e49iT^{2} \)
89 \( 1 + 1.06e25iT - 4.83e50T^{2} \)
97 \( 1 + (-5.76e25 + 5.76e25i)T - 4.52e51iT^{2} \)
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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

−16.79199384256341802116813877428, −15.62032719999225974879047390403, −14.27841879278944220095970176232, −12.06734872523326694818228018283, −10.13909662303580780043161448147, −8.190891145545300373992341131172, −7.61173468787828737573523698056, −4.43835548610039513463596633290, −3.25742576949895233323283624092, −0.41564062939525247314409141899, 1.49724487194866045434840191609, 2.62514240354047522934534826637, 5.22323586560877944130587922819, 7.72694416356045244474627586030, 8.750104582938770499626312158675, 10.91420760072057357087737146506, 12.24406265751558603067081389979, 14.35736985241553829622432876857, 15.40260588639157056716572067647, 18.07459581699773439291558661008

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