Properties

Label 2-5e2-1.1-c7-0-6
Degree $2$
Conductor $25$
Sign $1$
Analytic cond. $7.80962$
Root an. cond. $2.79457$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 20.2·2-s + 45.4·3-s + 281.·4-s + 920.·6-s − 1.36e3·7-s + 3.10e3·8-s − 118.·9-s − 1.01e3·11-s + 1.28e4·12-s + 3.64e3·13-s − 2.77e4·14-s + 2.68e4·16-s − 5.53e3·17-s − 2.40e3·18-s + 2.32e4·19-s − 6.23e4·21-s − 2.04e4·22-s + 9.63e4·23-s + 1.41e5·24-s + 7.37e4·26-s − 1.04e5·27-s − 3.85e5·28-s − 1.39e5·29-s − 2.08e5·31-s + 1.45e5·32-s − 4.60e4·33-s − 1.11e5·34-s + ⋯
L(s)  = 1  + 1.78·2-s + 0.972·3-s + 2.19·4-s + 1.73·6-s − 1.50·7-s + 2.14·8-s − 0.0544·9-s − 0.229·11-s + 2.13·12-s + 0.459·13-s − 2.70·14-s + 1.63·16-s − 0.273·17-s − 0.0973·18-s + 0.777·19-s − 1.46·21-s − 0.410·22-s + 1.65·23-s + 2.08·24-s + 0.822·26-s − 1.02·27-s − 3.32·28-s − 1.06·29-s − 1.25·31-s + 0.785·32-s − 0.223·33-s − 0.488·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $1$
Analytic conductor: \(7.80962\)
Root analytic conductor: \(2.79457\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(5.013878409\)
\(L(\frac12)\) \(\approx\) \(5.013878409\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 - 20.2T + 128T^{2} \)
3 \( 1 - 45.4T + 2.18e3T^{2} \)
7 \( 1 + 1.36e3T + 8.23e5T^{2} \)
11 \( 1 + 1.01e3T + 1.94e7T^{2} \)
13 \( 1 - 3.64e3T + 6.27e7T^{2} \)
17 \( 1 + 5.53e3T + 4.10e8T^{2} \)
19 \( 1 - 2.32e4T + 8.93e8T^{2} \)
23 \( 1 - 9.63e4T + 3.40e9T^{2} \)
29 \( 1 + 1.39e5T + 1.72e10T^{2} \)
31 \( 1 + 2.08e5T + 2.75e10T^{2} \)
37 \( 1 - 4.48e5T + 9.49e10T^{2} \)
41 \( 1 + 7.97e4T + 1.94e11T^{2} \)
43 \( 1 - 2.27e5T + 2.71e11T^{2} \)
47 \( 1 - 7.19e5T + 5.06e11T^{2} \)
53 \( 1 + 4.95e5T + 1.17e12T^{2} \)
59 \( 1 + 8.10e4T + 2.48e12T^{2} \)
61 \( 1 - 3.15e6T + 3.14e12T^{2} \)
67 \( 1 - 1.06e6T + 6.06e12T^{2} \)
71 \( 1 + 3.00e6T + 9.09e12T^{2} \)
73 \( 1 - 4.05e5T + 1.10e13T^{2} \)
79 \( 1 + 4.71e6T + 1.92e13T^{2} \)
83 \( 1 - 5.14e6T + 2.71e13T^{2} \)
89 \( 1 + 8.62e6T + 4.42e13T^{2} \)
97 \( 1 + 6.99e6T + 8.07e13T^{2} \)
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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

−15.51518151933933341231219317522, −14.55218128236128044839113501202, −13.35562242036575649682662563033, −12.86144526156551222849648386674, −11.20308232999855585134129311290, −9.281635666233924401419681334715, −7.15603504233198576598314518168, −5.70731033312829797868603045237, −3.67412521933221315571864002149, −2.73539118290749789693889009186, 2.73539118290749789693889009186, 3.67412521933221315571864002149, 5.70731033312829797868603045237, 7.15603504233198576598314518168, 9.281635666233924401419681334715, 11.20308232999855585134129311290, 12.86144526156551222849648386674, 13.35562242036575649682662563033, 14.55218128236128044839113501202, 15.51518151933933341231219317522

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