Properties

Degree $2$
Conductor $6025$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.95·2-s + 0.350·3-s + 1.83·4-s − 0.686·6-s + 1.14·7-s + 0.318·8-s − 2.87·9-s + 0.395·11-s + 0.643·12-s − 0.595·13-s − 2.25·14-s − 4.29·16-s − 7.79·17-s + 5.63·18-s + 3.25·19-s + 0.402·21-s − 0.774·22-s + 3.49·23-s + 0.111·24-s + 1.16·26-s − 2.05·27-s + 2.11·28-s + 6.23·29-s − 9.01·31-s + 7.78·32-s + 0.138·33-s + 15.2·34-s + ⋯
L(s)  = 1  − 1.38·2-s + 0.202·3-s + 0.918·4-s − 0.280·6-s + 0.434·7-s + 0.112·8-s − 0.959·9-s + 0.119·11-s + 0.185·12-s − 0.165·13-s − 0.601·14-s − 1.07·16-s − 1.88·17-s + 1.32·18-s + 0.746·19-s + 0.0878·21-s − 0.165·22-s + 0.729·23-s + 0.0228·24-s + 0.228·26-s − 0.396·27-s + 0.398·28-s + 1.15·29-s − 1.61·31-s + 1.37·32-s + 0.0241·33-s + 2.61·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{6025} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6025,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 - T \)
good2 \( 1 + 1.95T + 2T^{2} \)
3 \( 1 - 0.350T + 3T^{2} \)
7 \( 1 - 1.14T + 7T^{2} \)
11 \( 1 - 0.395T + 11T^{2} \)
13 \( 1 + 0.595T + 13T^{2} \)
17 \( 1 + 7.79T + 17T^{2} \)
19 \( 1 - 3.25T + 19T^{2} \)
23 \( 1 - 3.49T + 23T^{2} \)
29 \( 1 - 6.23T + 29T^{2} \)
31 \( 1 + 9.01T + 31T^{2} \)
37 \( 1 - 0.667T + 37T^{2} \)
41 \( 1 - 6.05T + 41T^{2} \)
43 \( 1 - 9.26T + 43T^{2} \)
47 \( 1 - 13.6T + 47T^{2} \)
53 \( 1 - 0.719T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 - 2.84T + 61T^{2} \)
67 \( 1 - 1.17T + 67T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 + 4.89T + 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 - 5.86T + 83T^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 - 0.175T + 97T^{2} \)
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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

−7.81171201515408972309841832360, −7.33113022177172149726804589371, −6.57654131161945059437880583741, −5.71154714674878836269980125836, −4.80988061072240494324622250920, −4.08260799986181477868457216642, −2.80609016031972207749190931404, −2.19228729560544830724831016844, −1.10411883823313101568396456016, 0, 1.10411883823313101568396456016, 2.19228729560544830724831016844, 2.80609016031972207749190931404, 4.08260799986181477868457216642, 4.80988061072240494324622250920, 5.71154714674878836269980125836, 6.57654131161945059437880583741, 7.33113022177172149726804589371, 7.81171201515408972309841832360

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