Properties

Degree $2$
Conductor $25$
Sign $-1$
Motivic weight $9$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 20.2·2-s − 30.5·3-s − 103.·4-s − 616.·6-s + 4.01e3·7-s − 1.24e4·8-s − 1.87e4·9-s − 4.21e4·11-s + 3.16e3·12-s − 1.23e5·13-s + 8.10e4·14-s − 1.98e5·16-s − 3.19e5·17-s − 3.78e5·18-s + 1.08e6·19-s − 1.22e5·21-s − 8.50e5·22-s − 1.50e6·23-s + 3.79e5·24-s − 2.49e6·26-s + 1.17e6·27-s − 4.16e5·28-s − 2.62e6·29-s + 3.27e6·31-s + 2.36e6·32-s + 1.28e6·33-s − 6.46e6·34-s + ⋯
L(s)  = 1  + 0.892·2-s − 0.217·3-s − 0.202·4-s − 0.194·6-s + 0.631·7-s − 1.07·8-s − 0.952·9-s − 0.867·11-s + 0.0441·12-s − 1.20·13-s + 0.563·14-s − 0.755·16-s − 0.929·17-s − 0.850·18-s + 1.91·19-s − 0.137·21-s − 0.774·22-s − 1.12·23-s + 0.233·24-s − 1.07·26-s + 0.424·27-s − 0.128·28-s − 0.688·29-s + 0.635·31-s + 0.399·32-s + 0.188·33-s − 0.829·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-1$
Motivic weight: \(9\)
Character: $\chi_{25} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 25,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 - 20.2T + 512T^{2} \)
3 \( 1 + 30.5T + 1.96e4T^{2} \)
7 \( 1 - 4.01e3T + 4.03e7T^{2} \)
11 \( 1 + 4.21e4T + 2.35e9T^{2} \)
13 \( 1 + 1.23e5T + 1.06e10T^{2} \)
17 \( 1 + 3.19e5T + 1.18e11T^{2} \)
19 \( 1 - 1.08e6T + 3.22e11T^{2} \)
23 \( 1 + 1.50e6T + 1.80e12T^{2} \)
29 \( 1 + 2.62e6T + 1.45e13T^{2} \)
31 \( 1 - 3.27e6T + 2.64e13T^{2} \)
37 \( 1 - 2.51e6T + 1.29e14T^{2} \)
41 \( 1 - 2.95e7T + 3.27e14T^{2} \)
43 \( 1 - 1.42e7T + 5.02e14T^{2} \)
47 \( 1 + 1.35e6T + 1.11e15T^{2} \)
53 \( 1 + 9.73e7T + 3.29e15T^{2} \)
59 \( 1 + 7.48e6T + 8.66e15T^{2} \)
61 \( 1 + 9.11e7T + 1.16e16T^{2} \)
67 \( 1 + 2.94e8T + 2.72e16T^{2} \)
71 \( 1 - 1.56e8T + 4.58e16T^{2} \)
73 \( 1 - 2.82e8T + 5.88e16T^{2} \)
79 \( 1 + 5.55e8T + 1.19e17T^{2} \)
83 \( 1 + 6.48e6T + 1.86e17T^{2} \)
89 \( 1 + 5.99e8T + 3.50e17T^{2} \)
97 \( 1 + 9.25e8T + 7.60e17T^{2} \)
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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

−14.54192802842692284546358285298, −13.75364134509268459491960891098, −12.35701020613872017876767637838, −11.30244727891195735897360890196, −9.475566648017787305391727405541, −7.84078528087851967200430637627, −5.73105075948872739414964611968, −4.69893143914022749985463743303, −2.76087902285144678126819856350, 0, 2.76087902285144678126819856350, 4.69893143914022749985463743303, 5.73105075948872739414964611968, 7.84078528087851967200430637627, 9.475566648017787305391727405541, 11.30244727891195735897360890196, 12.35701020613872017876767637838, 13.75364134509268459491960891098, 14.54192802842692284546358285298

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