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Properties

Label 2-5-5.3-c26-0-1
Degree $2$
Conductor $5$
Sign $0.808 + 0.587i$
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  + (584. − 584. i)2-s + (−2.17e6 − 2.17e6i)3-s + 6.64e7i·4-s + (−1.21e9 − 9.14e7i)5-s − 2.53e9·6-s + (−4.74e10 + 4.74e10i)7-s + (7.81e10 + 7.81e10i)8-s + 6.88e12i·9-s + (−7.65e11 + 6.58e11i)10-s − 1.55e13·11-s + (1.44e14 − 1.44e14i)12-s + (−2.89e14 − 2.89e14i)13-s + 5.55e13i·14-s + (2.44e15 + 2.84e15i)15-s − 4.36e15·16-s + (1.79e15 − 1.79e15i)17-s + ⋯
L(s)  = 1  + (0.0713 − 0.0713i)2-s + (−1.36 − 1.36i)3-s + 0.989i·4-s + (−0.997 − 0.0749i)5-s − 0.194·6-s + (−0.490 + 0.490i)7-s + (0.142 + 0.142i)8-s + 2.70i·9-s + (−0.0765 + 0.0658i)10-s − 0.451·11-s + (1.34 − 1.34i)12-s + (−0.957 − 0.957i)13-s + 0.0699i·14-s + (1.25 + 1.46i)15-s − 0.969·16-s + (0.180 − 0.180i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.587i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (0.808 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.808 + 0.587i$
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: \((2,\ 5,\ (\ :13),\ 0.808 + 0.587i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(0.453175 - 0.147306i\)
\(L(\frac12)\) \(\approx\) \(0.453175 - 0.147306i\)
\(L(14)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.21e9 + 9.14e7i)T \)
good2 \( 1 + (-584. + 584. i)T - 6.71e7iT^{2} \)
3 \( 1 + (2.17e6 + 2.17e6i)T + 2.54e12iT^{2} \)
7 \( 1 + (4.74e10 - 4.74e10i)T - 9.38e21iT^{2} \)
11 \( 1 + 1.55e13T + 1.19e27T^{2} \)
13 \( 1 + (2.89e14 + 2.89e14i)T + 9.17e28iT^{2} \)
17 \( 1 + (-1.79e15 + 1.79e15i)T - 9.81e31iT^{2} \)
19 \( 1 + 3.35e16iT - 1.76e33T^{2} \)
23 \( 1 + (-9.60e16 - 9.60e16i)T + 2.54e35iT^{2} \)
29 \( 1 - 5.76e18iT - 1.05e38T^{2} \)
31 \( 1 + 2.05e18T + 5.96e38T^{2} \)
37 \( 1 + (-1.41e20 + 1.41e20i)T - 5.93e40iT^{2} \)
41 \( 1 + 1.34e21T + 8.55e41T^{2} \)
43 \( 1 + (-1.49e21 - 1.49e21i)T + 2.95e42iT^{2} \)
47 \( 1 + (-5.76e20 + 5.76e20i)T - 2.98e43iT^{2} \)
53 \( 1 + (1.74e22 + 1.74e22i)T + 6.77e44iT^{2} \)
59 \( 1 - 7.32e22iT - 1.10e46T^{2} \)
61 \( 1 - 1.97e23T + 2.62e46T^{2} \)
67 \( 1 + (-2.61e23 + 2.61e23i)T - 3.00e47iT^{2} \)
71 \( 1 - 1.02e24T + 1.35e48T^{2} \)
73 \( 1 + (-9.63e23 - 9.63e23i)T + 2.79e48iT^{2} \)
79 \( 1 - 4.00e24iT - 2.17e49T^{2} \)
83 \( 1 + (-4.36e24 - 4.36e24i)T + 7.87e49iT^{2} \)
89 \( 1 + 3.55e25iT - 4.83e50T^{2} \)
97 \( 1 + (-1.59e25 + 1.59e25i)T - 4.52e51iT^{2} \)
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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

−17.19669059296323445845825788710, −15.95052097492149952231331443607, −12.95300790936518748268588251964, −12.32916542940179409652096927457, −11.20248093458834996224812848028, −7.963175148885577867997237677516, −6.99316763063009527774771999543, −5.09573380769497898790553778682, −2.69762474275424915018665176903, −0.45080107169017461834866313912, 0.50310162283713715837617846672, 3.96815288937264213764959903124, 5.07559621740435357969692616745, 6.63001193171396512300046751556, 9.693305026624016467087717435270, 10.69425465646737387437769769960, 11.94435861914640236793882362537, 14.77473465647677145108226019242, 15.86589962937973397950478781746, 16.86218138336863670019827202727

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