Properties

Label 2-5-1.1-c13-0-4
Degree $2$
Conductor $5$
Sign $-1$
Analytic cond. $5.36154$
Root an. cond. $2.31550$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 49.3·2-s − 682.·3-s − 5.75e3·4-s − 1.56e4·5-s − 3.36e4·6-s − 1.61e5·7-s − 6.88e5·8-s − 1.12e6·9-s − 7.71e5·10-s + 5.82e6·11-s + 3.92e6·12-s + 1.51e7·13-s − 7.99e6·14-s + 1.06e7·15-s + 1.31e7·16-s − 9.09e7·17-s − 5.57e7·18-s − 1.47e8·19-s + 8.99e7·20-s + 1.10e8·21-s + 2.87e8·22-s − 5.39e8·23-s + 4.69e8·24-s + 2.44e8·25-s + 7.48e8·26-s + 1.85e9·27-s + 9.32e8·28-s + ⋯
L(s)  = 1  + 0.545·2-s − 0.540·3-s − 0.702·4-s − 0.447·5-s − 0.294·6-s − 0.520·7-s − 0.928·8-s − 0.708·9-s − 0.243·10-s + 0.991·11-s + 0.379·12-s + 0.871·13-s − 0.283·14-s + 0.241·15-s + 0.196·16-s − 0.913·17-s − 0.386·18-s − 0.720·19-s + 0.314·20-s + 0.281·21-s + 0.540·22-s − 0.760·23-s + 0.501·24-s + 0.199·25-s + 0.475·26-s + 0.922·27-s + 0.365·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-1$
Analytic conductor: \(5.36154\)
Root analytic conductor: \(2.31550\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 1.56e4T \)
good2 \( 1 - 49.3T + 8.19e3T^{2} \)
3 \( 1 + 682.T + 1.59e6T^{2} \)
7 \( 1 + 1.61e5T + 9.68e10T^{2} \)
11 \( 1 - 5.82e6T + 3.45e13T^{2} \)
13 \( 1 - 1.51e7T + 3.02e14T^{2} \)
17 \( 1 + 9.09e7T + 9.90e15T^{2} \)
19 \( 1 + 1.47e8T + 4.20e16T^{2} \)
23 \( 1 + 5.39e8T + 5.04e17T^{2} \)
29 \( 1 + 5.51e9T + 1.02e19T^{2} \)
31 \( 1 + 7.83e9T + 2.44e19T^{2} \)
37 \( 1 - 2.87e10T + 2.43e20T^{2} \)
41 \( 1 - 3.61e9T + 9.25e20T^{2} \)
43 \( 1 + 3.01e10T + 1.71e21T^{2} \)
47 \( 1 - 1.86e10T + 5.46e21T^{2} \)
53 \( 1 - 2.06e11T + 2.60e22T^{2} \)
59 \( 1 - 3.80e11T + 1.04e23T^{2} \)
61 \( 1 + 3.75e11T + 1.61e23T^{2} \)
67 \( 1 + 1.19e12T + 5.48e23T^{2} \)
71 \( 1 - 6.59e11T + 1.16e24T^{2} \)
73 \( 1 - 3.87e11T + 1.67e24T^{2} \)
79 \( 1 + 3.71e12T + 4.66e24T^{2} \)
83 \( 1 + 7.97e11T + 8.87e24T^{2} \)
89 \( 1 + 2.74e11T + 2.19e25T^{2} \)
97 \( 1 - 1.19e13T + 6.73e25T^{2} \)
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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

−19.94221346413179917034256244813, −18.23499171556847430907301785184, −16.67736664763878180986342979023, −14.73718640817609947753306396156, −13.06090388683822756072475660153, −11.42476079684535018007315698505, −8.941977158864779004370207755072, −6.04262873600789091902022133204, −3.91275303115995055431909322721, 0, 3.91275303115995055431909322721, 6.04262873600789091902022133204, 8.941977158864779004370207755072, 11.42476079684535018007315698505, 13.06090388683822756072475660153, 14.73718640817609947753306396156, 16.67736664763878180986342979023, 18.23499171556847430907301785184, 19.94221346413179917034256244813

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