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  − 2.68·2-s + 2.58·3-s + 5.21·4-s − 6.93·6-s + 1.24·7-s − 8.64·8-s + 3.66·9-s − 5.71·11-s + 13.4·12-s − 3.28·13-s − 3.35·14-s + 12.7·16-s + 5.29·17-s − 9.83·18-s + 6.14·19-s + 3.22·21-s + 15.3·22-s + 2.22·23-s − 22.3·24-s + 8.82·26-s + 1.70·27-s + 6.52·28-s − 5.09·29-s − 4.50·31-s − 17.0·32-s − 14.7·33-s − 14.2·34-s + ⋯
L(s)  = 1  − 1.89·2-s + 1.48·3-s + 2.60·4-s − 2.83·6-s + 0.472·7-s − 3.05·8-s + 1.22·9-s − 1.72·11-s + 3.88·12-s − 0.910·13-s − 0.897·14-s + 3.19·16-s + 1.28·17-s − 2.31·18-s + 1.41·19-s + 0.703·21-s + 3.27·22-s + 0.463·23-s − 4.55·24-s + 1.72·26-s + 0.327·27-s + 1.23·28-s − 0.945·29-s − 0.809·31-s − 3.02·32-s − 2.56·33-s − 2.43·34-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 \( 1 + 2.68T + 2T^{2} \)
3 \( 1 - 2.58T + 3T^{2} \)
7 \( 1 - 1.24T + 7T^{2} \)
11 \( 1 + 5.71T + 11T^{2} \)
13 \( 1 + 3.28T + 13T^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 - 6.14T + 19T^{2} \)
23 \( 1 - 2.22T + 23T^{2} \)
29 \( 1 + 5.09T + 29T^{2} \)
31 \( 1 + 4.50T + 31T^{2} \)
37 \( 1 + 4.88T + 37T^{2} \)
41 \( 1 + 4.77T + 41T^{2} \)
43 \( 1 - 2.72T + 43T^{2} \)
47 \( 1 + 2.52T + 47T^{2} \)
53 \( 1 + 7.67T + 53T^{2} \)
59 \( 1 + 8.39T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 - 15.2T + 67T^{2} \)
71 \( 1 + 6.27T + 71T^{2} \)
73 \( 1 + 5.27T + 73T^{2} \)
79 \( 1 - 9.38T + 79T^{2} \)
83 \( 1 + 2.13T + 83T^{2} \)
89 \( 1 - 8.11T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−7.80857423010491642292126965968, −7.60989329556491659121381056283, −6.99398198318643013800642569885, −5.64754906672283468141652913309, −5.02621259385270195957859394232, −3.30094178785788899945096661416, −2.99912611380657027884427180006, −2.11759563876215184235691672836, −1.42403811165084506519058103549, 0, 1.42403811165084506519058103549, 2.11759563876215184235691672836, 2.99912611380657027884427180006, 3.30094178785788899945096661416, 5.02621259385270195957859394232, 5.64754906672283468141652913309, 6.99398198318643013800642569885, 7.60989329556491659121381056283, 7.80857423010491642292126965968

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