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  − 21.4·2-s + 210.·3-s − 53.2·4-s − 4.50e3·6-s − 9.90e3·7-s + 1.21e4·8-s + 2.44e4·9-s + 3.64e4·11-s − 1.11e4·12-s − 1.64e5·13-s + 2.12e5·14-s − 2.32e5·16-s − 8.23e4·17-s − 5.24e5·18-s − 6.09e5·19-s − 2.08e6·21-s − 7.80e5·22-s − 1.88e6·23-s + 2.54e6·24-s + 3.53e6·26-s + 1.01e6·27-s + 5.27e5·28-s + 3.39e5·29-s + 5.47e5·31-s − 1.22e6·32-s + 7.66e6·33-s + 1.76e6·34-s + ⋯
L(s)  = 1  − 0.946·2-s + 1.49·3-s − 0.103·4-s − 1.41·6-s − 1.55·7-s + 1.04·8-s + 1.24·9-s + 0.750·11-s − 0.155·12-s − 1.60·13-s + 1.47·14-s − 0.885·16-s − 0.239·17-s − 1.17·18-s − 1.07·19-s − 2.33·21-s − 0.710·22-s − 1.40·23-s + 1.56·24-s + 1.51·26-s + 0.365·27-s + 0.162·28-s + 0.0890·29-s + 0.106·31-s − 0.207·32-s + 1.12·33-s + 0.226·34-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 \( 1 + 21.4T + 512T^{2} \)
3 \( 1 - 210.T + 1.96e4T^{2} \)
7 \( 1 + 9.90e3T + 4.03e7T^{2} \)
11 \( 1 - 3.64e4T + 2.35e9T^{2} \)
13 \( 1 + 1.64e5T + 1.06e10T^{2} \)
17 \( 1 + 8.23e4T + 1.18e11T^{2} \)
19 \( 1 + 6.09e5T + 3.22e11T^{2} \)
23 \( 1 + 1.88e6T + 1.80e12T^{2} \)
29 \( 1 - 3.39e5T + 1.45e13T^{2} \)
31 \( 1 - 5.47e5T + 2.64e13T^{2} \)
37 \( 1 + 5.25e6T + 1.29e14T^{2} \)
41 \( 1 - 2.05e6T + 3.27e14T^{2} \)
43 \( 1 - 6.76e6T + 5.02e14T^{2} \)
47 \( 1 + 3.15e7T + 1.11e15T^{2} \)
53 \( 1 - 4.89e7T + 3.29e15T^{2} \)
59 \( 1 - 8.77e7T + 8.66e15T^{2} \)
61 \( 1 - 3.84e7T + 1.16e16T^{2} \)
67 \( 1 - 1.36e8T + 2.72e16T^{2} \)
71 \( 1 - 3.49e8T + 4.58e16T^{2} \)
73 \( 1 + 1.61e8T + 5.88e16T^{2} \)
79 \( 1 + 1.26e8T + 1.19e17T^{2} \)
83 \( 1 - 2.87e8T + 1.86e17T^{2} \)
89 \( 1 + 5.63e8T + 3.50e17T^{2} \)
97 \( 1 - 4.71e8T + 7.60e17T^{2} \)
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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

−14.79338497269899342508324449967, −13.70917626106507298691176968364, −12.55770890990143471509376138485, −10.00152887582417565495041228708, −9.441403860475069735441949996622, −8.319240754853765611889985219103, −6.93609597030698285625299696812, −3.95119921526953394778238225382, −2.29197459251356216679271108537, 0, 2.29197459251356216679271108537, 3.95119921526953394778238225382, 6.93609597030698285625299696812, 8.319240754853765611889985219103, 9.441403860475069735441949996622, 10.00152887582417565495041228708, 12.55770890990143471509376138485, 13.70917626106507298691176968364, 14.79338497269899342508324449967

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