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

−7.64998583217705747586637748749, −6.89721052912210181724836417347, −6.19992930741870432539649440424, −5.43304337921537415812280909747, −4.73871652739146237421866163568, −3.95041342182080236036572896887, −3.20866189759665950582425431558, −2.79559717060908412677943013539, −1.83436493625413832625341570789, 0, 1.83436493625413832625341570789, 2.79559717060908412677943013539, 3.20866189759665950582425431558, 3.95041342182080236036572896887, 4.73871652739146237421866163568, 5.43304337921537415812280909747, 6.19992930741870432539649440424, 6.89721052912210181724836417347, 7.64998583217705747586637748749

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