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Properties

Label 2-5-5.2-c26-0-9
Degree $2$
Conductor $5$
Sign $-0.914 + 0.405i$
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  + (−9.19e3 − 9.19e3i)2-s + (1.92e6 − 1.92e6i)3-s + 1.02e8i·4-s + (1.20e9 + 1.65e8i)5-s − 3.53e10·6-s + (2.78e10 + 2.78e10i)7-s + (3.22e11 − 3.22e11i)8-s − 4.84e12i·9-s + (−9.60e12 − 1.26e13i)10-s + 5.43e13·11-s + (1.96e14 + 1.96e14i)12-s + (9.46e13 − 9.46e13i)13-s − 5.12e14i·14-s + (2.64e15 − 2.00e15i)15-s + 9.22e14·16-s + (−9.19e15 − 9.19e15i)17-s + ⋯
L(s)  = 1  + (−1.12 − 1.12i)2-s + (1.20 − 1.20i)3-s + 1.52i·4-s + (0.990 + 0.135i)5-s − 2.70·6-s + (0.287 + 0.287i)7-s + (0.586 − 0.586i)8-s − 1.90i·9-s + (−0.960 − 1.26i)10-s + 1.57·11-s + (1.83 + 1.83i)12-s + (0.312 − 0.312i)13-s − 0.645i·14-s + (1.35 − 1.03i)15-s + 0.204·16-s + (−0.927 − 0.927i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.405i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (-0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.914 + 0.405i$
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: \((2,\ 5,\ (\ :13),\ -0.914 + 0.405i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(0.428294 - 2.02087i\)
\(L(\frac12)\) \(\approx\) \(0.428294 - 2.02087i\)
\(L(14)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.20e9 - 1.65e8i)T \)
good2 \( 1 + (9.19e3 + 9.19e3i)T + 6.71e7iT^{2} \)
3 \( 1 + (-1.92e6 + 1.92e6i)T - 2.54e12iT^{2} \)
7 \( 1 + (-2.78e10 - 2.78e10i)T + 9.38e21iT^{2} \)
11 \( 1 - 5.43e13T + 1.19e27T^{2} \)
13 \( 1 + (-9.46e13 + 9.46e13i)T - 9.17e28iT^{2} \)
17 \( 1 + (9.19e15 + 9.19e15i)T + 9.81e31iT^{2} \)
19 \( 1 + 5.32e16iT - 1.76e33T^{2} \)
23 \( 1 + (1.98e17 - 1.98e17i)T - 2.54e35iT^{2} \)
29 \( 1 - 1.15e19iT - 1.05e38T^{2} \)
31 \( 1 - 7.75e18T + 5.96e38T^{2} \)
37 \( 1 + (1.01e19 + 1.01e19i)T + 5.93e40iT^{2} \)
41 \( 1 - 6.54e20T + 8.55e41T^{2} \)
43 \( 1 + (1.72e21 - 1.72e21i)T - 2.95e42iT^{2} \)
47 \( 1 + (1.35e21 + 1.35e21i)T + 2.98e43iT^{2} \)
53 \( 1 + (1.28e22 - 1.28e22i)T - 6.77e44iT^{2} \)
59 \( 1 + 5.88e22iT - 1.10e46T^{2} \)
61 \( 1 - 1.54e23T + 2.62e46T^{2} \)
67 \( 1 + (-1.66e23 - 1.66e23i)T + 3.00e47iT^{2} \)
71 \( 1 + 1.12e24T + 1.35e48T^{2} \)
73 \( 1 + (9.51e23 - 9.51e23i)T - 2.79e48iT^{2} \)
79 \( 1 - 2.79e24iT - 2.17e49T^{2} \)
83 \( 1 + (1.23e24 - 1.23e24i)T - 7.87e49iT^{2} \)
89 \( 1 + 3.17e25iT - 4.83e50T^{2} \)
97 \( 1 + (-1.53e25 - 1.53e25i)T + 4.52e51iT^{2} \)
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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

−17.57889151786896980835190747734, −14.40255779088064802277586137435, −13.10401981886047767619927684387, −11.52925266375372462676173561834, −9.396921817477673386269083628174, −8.671012378745030534012117534141, −6.83239295874323833059090185741, −2.97583612900847433462998716743, −1.91979604685023932938756998030, −1.02039838726054914770343248181, 1.66742448203099728285887710254, 4.06344235935021773137705220051, 6.26759195710400626718047411640, 8.342566512128349886133088057760, 9.245975535976416360289179805130, 10.24399047890659942798098402602, 14.04941511408785290536241150668, 14.92109997383706005189796443695, 16.40825536028158769898642385687, 17.39888506378641647374714316406

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