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  − 31.7·2-s − 268.·3-s + 498.·4-s + 8.53e3·6-s + 637.·7-s + 440.·8-s + 5.24e4·9-s − 4.90e4·11-s − 1.33e5·12-s + 7.27e4·13-s − 2.02e4·14-s − 2.69e5·16-s + 6.73e4·17-s − 1.66e6·18-s + 3.41e5·19-s − 1.71e5·21-s + 1.55e6·22-s − 1.34e5·23-s − 1.18e5·24-s − 2.31e6·26-s − 8.81e6·27-s + 3.17e5·28-s + 4.45e6·29-s + 4.56e5·31-s + 8.32e6·32-s + 1.31e7·33-s − 2.13e6·34-s + ⋯
L(s)  = 1  − 1.40·2-s − 1.91·3-s + 0.972·4-s + 2.68·6-s + 0.100·7-s + 0.0380·8-s + 2.66·9-s − 1.00·11-s − 1.86·12-s + 0.706·13-s − 0.140·14-s − 1.02·16-s + 0.195·17-s − 3.74·18-s + 0.600·19-s − 0.192·21-s + 1.41·22-s − 0.100·23-s − 0.0727·24-s − 0.991·26-s − 3.19·27-s + 0.0975·28-s + 1.17·29-s + 0.0887·31-s + 1.40·32-s + 1.93·33-s − 0.274·34-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 \( 1 + 31.7T + 512T^{2} \)
3 \( 1 + 268.T + 1.96e4T^{2} \)
7 \( 1 - 637.T + 4.03e7T^{2} \)
11 \( 1 + 4.90e4T + 2.35e9T^{2} \)
13 \( 1 - 7.27e4T + 1.06e10T^{2} \)
17 \( 1 - 6.73e4T + 1.18e11T^{2} \)
19 \( 1 - 3.41e5T + 3.22e11T^{2} \)
23 \( 1 + 1.34e5T + 1.80e12T^{2} \)
29 \( 1 - 4.45e6T + 1.45e13T^{2} \)
31 \( 1 - 4.56e5T + 2.64e13T^{2} \)
37 \( 1 - 1.30e7T + 1.29e14T^{2} \)
41 \( 1 + 2.56e7T + 3.27e14T^{2} \)
43 \( 1 - 3.42e6T + 5.02e14T^{2} \)
47 \( 1 + 3.39e7T + 1.11e15T^{2} \)
53 \( 1 + 8.42e7T + 3.29e15T^{2} \)
59 \( 1 + 7.46e7T + 8.66e15T^{2} \)
61 \( 1 - 1.78e8T + 1.16e16T^{2} \)
67 \( 1 - 6.94e7T + 2.72e16T^{2} \)
71 \( 1 + 2.07e8T + 4.58e16T^{2} \)
73 \( 1 + 3.02e8T + 5.88e16T^{2} \)
79 \( 1 - 3.72e8T + 1.19e17T^{2} \)
83 \( 1 - 4.50e8T + 1.86e17T^{2} \)
89 \( 1 - 5.82e7T + 3.50e17T^{2} \)
97 \( 1 - 7.85e8T + 7.60e17T^{2} \)
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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

−15.81668678859340636348568832121, −13.16619608604816373812186690775, −11.68169259467473820630641620589, −10.73460499143510124342817246010, −9.855387815386657612873864730795, −7.903825817238894517680062799071, −6.48045144075158179156089249643, −4.94605313112059336701637850812, −1.23343417415332397762648624664, 0, 1.23343417415332397762648624664, 4.94605313112059336701637850812, 6.48045144075158179156089249643, 7.903825817238894517680062799071, 9.855387815386657612873864730795, 10.73460499143510124342817246010, 11.68169259467473820630641620589, 13.16619608604816373812186690775, 15.81668678859340636348568832121

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