Properties

Label 2-5-1.1-c27-0-0
Degree $2$
Conductor $5$
Sign $1$
Analytic cond. $23.0927$
Root an. cond. $4.80549$
Motivic weight $27$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.16e3·2-s − 2.41e6·3-s − 1.32e8·4-s − 1.22e9·5-s − 2.81e9·6-s − 4.04e11·7-s − 3.11e11·8-s − 1.78e12·9-s − 1.42e12·10-s − 7.68e13·11-s + 3.20e14·12-s + 4.89e13·13-s − 4.71e14·14-s + 2.94e15·15-s + 1.74e16·16-s − 5.14e16·17-s − 2.08e15·18-s − 2.66e16·19-s + 1.62e17·20-s + 9.76e17·21-s − 8.94e16·22-s + 1.01e18·23-s + 7.51e17·24-s + 1.49e18·25-s + 5.70e16·26-s + 2.27e19·27-s + 5.37e19·28-s + ⋯
L(s)  = 1  + 0.100·2-s − 0.874·3-s − 0.989·4-s − 0.447·5-s − 0.0879·6-s − 1.57·7-s − 0.200·8-s − 0.234·9-s − 0.0449·10-s − 0.670·11-s + 0.865·12-s + 0.0448·13-s − 0.158·14-s + 0.391·15-s + 0.969·16-s − 1.25·17-s − 0.0236·18-s − 0.145·19-s + 0.442·20-s + 1.37·21-s − 0.0674·22-s + 0.418·23-s + 0.175·24-s + 0.199·25-s + 0.00450·26-s + 1.08·27-s + 1.56·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $1$
Analytic conductor: \(23.0927\)
Root analytic conductor: \(4.80549\)
Motivic weight: \(27\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :27/2),\ 1)\)

Particular Values

\(L(14)\) \(\approx\) \(0.1369357034\)
\(L(\frac12)\) \(\approx\) \(0.1369357034\)
\(L(\frac{29}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 1.22e9T \)
good2 \( 1 - 1.16e3T + 1.34e8T^{2} \)
3 \( 1 + 2.41e6T + 7.62e12T^{2} \)
7 \( 1 + 4.04e11T + 6.57e22T^{2} \)
11 \( 1 + 7.68e13T + 1.31e28T^{2} \)
13 \( 1 - 4.89e13T + 1.19e30T^{2} \)
17 \( 1 + 5.14e16T + 1.66e33T^{2} \)
19 \( 1 + 2.66e16T + 3.36e34T^{2} \)
23 \( 1 - 1.01e18T + 5.84e36T^{2} \)
29 \( 1 - 7.39e19T + 3.05e39T^{2} \)
31 \( 1 + 1.35e20T + 1.84e40T^{2} \)
37 \( 1 + 2.44e21T + 2.19e42T^{2} \)
41 \( 1 + 1.08e22T + 3.50e43T^{2} \)
43 \( 1 + 1.14e22T + 1.26e44T^{2} \)
47 \( 1 - 2.01e22T + 1.40e45T^{2} \)
53 \( 1 + 8.85e22T + 3.59e46T^{2} \)
59 \( 1 + 8.13e22T + 6.50e47T^{2} \)
61 \( 1 + 2.23e24T + 1.59e48T^{2} \)
67 \( 1 + 1.42e24T + 2.01e49T^{2} \)
71 \( 1 - 1.09e25T + 9.63e49T^{2} \)
73 \( 1 + 4.29e24T + 2.04e50T^{2} \)
79 \( 1 - 7.60e25T + 1.72e51T^{2} \)
83 \( 1 + 1.10e26T + 6.53e51T^{2} \)
89 \( 1 - 2.32e26T + 4.30e52T^{2} \)
97 \( 1 - 3.39e26T + 4.39e53T^{2} \)
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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

−17.00000596440694158471306248905, −15.62940267759252829086655630380, −13.48262230219382330821731180807, −12.31147538938884048157830860682, −10.44707828373400358568430028028, −8.809216607946195055687621343417, −6.53263312679586192716342819317, −5.00295508072671672356676307303, −3.29297946767003338143336366854, −0.24077449953947311192868358578, 0.24077449953947311192868358578, 3.29297946767003338143336366854, 5.00295508072671672356676307303, 6.53263312679586192716342819317, 8.809216607946195055687621343417, 10.44707828373400358568430028028, 12.31147538938884048157830860682, 13.48262230219382330821731180807, 15.62940267759252829086655630380, 17.00000596440694158471306248905

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