Properties

Label 2-5e2-1.1-c35-0-25
Degree $2$
Conductor $25$
Sign $1$
Analytic cond. $193.987$
Root an. cond. $13.9279$
Motivic weight $35$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.65e5·2-s + 3.45e8·3-s − 7.09e9·4-s + 5.71e13·6-s + 1.25e14·7-s − 6.84e15·8-s + 6.96e16·9-s − 1.70e18·11-s − 2.45e18·12-s + 4.94e19·13-s + 2.06e19·14-s − 8.86e20·16-s − 1.32e21·17-s + 1.14e22·18-s + 3.94e21·19-s + 4.32e22·21-s − 2.82e23·22-s + 3.48e23·23-s − 2.36e24·24-s + 8.17e24·26-s + 6.77e24·27-s − 8.88e23·28-s + 3.21e25·29-s + 3.41e25·31-s + 8.88e25·32-s − 5.90e26·33-s − 2.18e26·34-s + ⋯
L(s)  = 1  + 0.890·2-s + 1.54·3-s − 0.206·4-s + 1.37·6-s + 0.203·7-s − 1.07·8-s + 1.39·9-s − 1.01·11-s − 0.319·12-s + 1.58·13-s + 0.181·14-s − 0.750·16-s − 0.387·17-s + 1.23·18-s + 0.165·19-s + 0.314·21-s − 0.907·22-s + 0.515·23-s − 1.66·24-s + 1.41·26-s + 0.605·27-s − 0.0420·28-s + 0.822·29-s + 0.271·31-s + 0.406·32-s − 1.57·33-s − 0.344·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+35/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $1$
Analytic conductor: \(193.987\)
Root analytic conductor: \(13.9279\)
Motivic weight: \(35\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :35/2),\ 1)\)

Particular Values

\(L(18)\) \(\approx\) \(6.143734727\)
\(L(\frac12)\) \(\approx\) \(6.143734727\)
\(L(\frac{37}{2})\) 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 \( 1 - 1.65e5T + 3.43e10T^{2} \)
3 \( 1 - 3.45e8T + 5.00e16T^{2} \)
7 \( 1 - 1.25e14T + 3.78e29T^{2} \)
11 \( 1 + 1.70e18T + 2.81e36T^{2} \)
13 \( 1 - 4.94e19T + 9.72e38T^{2} \)
17 \( 1 + 1.32e21T + 1.16e43T^{2} \)
19 \( 1 - 3.94e21T + 5.70e44T^{2} \)
23 \( 1 - 3.48e23T + 4.57e47T^{2} \)
29 \( 1 - 3.21e25T + 1.52e51T^{2} \)
31 \( 1 - 3.41e25T + 1.57e52T^{2} \)
37 \( 1 - 4.03e27T + 7.71e54T^{2} \)
41 \( 1 - 8.65e27T + 2.80e56T^{2} \)
43 \( 1 + 9.89e26T + 1.48e57T^{2} \)
47 \( 1 + 1.95e29T + 3.33e58T^{2} \)
53 \( 1 - 9.96e29T + 2.23e60T^{2} \)
59 \( 1 - 3.91e30T + 9.54e61T^{2} \)
61 \( 1 - 7.64e29T + 3.06e62T^{2} \)
67 \( 1 - 1.64e32T + 8.17e63T^{2} \)
71 \( 1 - 7.65e31T + 6.22e64T^{2} \)
73 \( 1 - 7.08e32T + 1.64e65T^{2} \)
79 \( 1 - 2.34e33T + 2.61e66T^{2} \)
83 \( 1 + 5.18e33T + 1.47e67T^{2} \)
89 \( 1 - 1.44e34T + 1.69e68T^{2} \)
97 \( 1 - 2.87e34T + 3.44e69T^{2} \)
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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

−11.17782066067939378478189742571, −9.679014956705924170207484481947, −8.627218915237620246765424484494, −7.992723638347004197903551409577, −6.37235819100925621430252789213, −5.03344343462385709944499380133, −3.95041407355108660982121978600, −3.14708861949768754444563881321, −2.29655612935823461598146764275, −0.856631729584782953895287482152, 0.856631729584782953895287482152, 2.29655612935823461598146764275, 3.14708861949768754444563881321, 3.95041407355108660982121978600, 5.03344343462385709944499380133, 6.37235819100925621430252789213, 7.992723638347004197903551409577, 8.627218915237620246765424484494, 9.679014956705924170207484481947, 11.17782066067939378478189742571

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