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  − 1.84·2-s − 1.69·3-s + 1.39·4-s + 3.12·6-s + 1.54·7-s + 1.11·8-s − 0.120·9-s + 2.53·11-s − 2.36·12-s + 6.36·13-s − 2.83·14-s − 4.84·16-s − 3.54·17-s + 0.222·18-s + 0.713·19-s − 2.61·21-s − 4.67·22-s + 1.66·23-s − 1.89·24-s − 11.7·26-s + 5.29·27-s + 2.14·28-s − 8.23·29-s + 10.5·31-s + 6.69·32-s − 4.30·33-s + 6.52·34-s + ⋯
L(s)  = 1  − 1.30·2-s − 0.979·3-s + 0.696·4-s + 1.27·6-s + 0.582·7-s + 0.395·8-s − 0.0402·9-s + 0.764·11-s − 0.682·12-s + 1.76·13-s − 0.758·14-s − 1.21·16-s − 0.859·17-s + 0.0524·18-s + 0.163·19-s − 0.570·21-s − 0.995·22-s + 0.347·23-s − 0.387·24-s − 2.30·26-s + 1.01·27-s + 0.405·28-s − 1.53·29-s + 1.89·31-s + 1.18·32-s − 0.749·33-s + 1.11·34-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 \( 1 + 1.84T + 2T^{2} \)
3 \( 1 + 1.69T + 3T^{2} \)
7 \( 1 - 1.54T + 7T^{2} \)
11 \( 1 - 2.53T + 11T^{2} \)
13 \( 1 - 6.36T + 13T^{2} \)
17 \( 1 + 3.54T + 17T^{2} \)
19 \( 1 - 0.713T + 19T^{2} \)
23 \( 1 - 1.66T + 23T^{2} \)
29 \( 1 + 8.23T + 29T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 + 0.496T + 37T^{2} \)
41 \( 1 + 8.92T + 41T^{2} \)
43 \( 1 - 1.50T + 43T^{2} \)
47 \( 1 - 6.94T + 47T^{2} \)
53 \( 1 + 9.87T + 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 + 9.13T + 61T^{2} \)
67 \( 1 + 1.85T + 67T^{2} \)
71 \( 1 + 9.47T + 71T^{2} \)
73 \( 1 + 8.88T + 73T^{2} \)
79 \( 1 + 6.12T + 79T^{2} \)
83 \( 1 + 0.835T + 83T^{2} \)
89 \( 1 - 2.27T + 89T^{2} \)
97 \( 1 + 7.56T + 97T^{2} \)
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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

−7.908995054557943175255057510357, −7.05342085290604124084497256502, −6.34311510421423605459782481626, −5.88914637037995298383248560758, −4.80625571715996467640406049261, −4.25404461943419426994134581310, −3.11577740386111299472060235934, −1.68671407914604073859012630061, −1.15468115527073731507429438315, 0, 1.15468115527073731507429438315, 1.68671407914604073859012630061, 3.11577740386111299472060235934, 4.25404461943419426994134581310, 4.80625571715996467640406049261, 5.88914637037995298383248560758, 6.34311510421423605459782481626, 7.05342085290604124084497256502, 7.908995054557943175255057510357

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