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  + 0.260·2-s − 1.85·3-s − 1.93·4-s − 0.484·6-s − 1.27·7-s − 1.02·8-s + 0.456·9-s − 5.63·11-s + 3.59·12-s − 6.25·13-s − 0.332·14-s + 3.59·16-s + 0.195·17-s + 0.119·18-s + 3.74·19-s + 2.36·21-s − 1.46·22-s + 8.06·23-s + 1.90·24-s − 1.62·26-s + 4.72·27-s + 2.46·28-s + 6.42·29-s − 10.3·31-s + 2.98·32-s + 10.4·33-s + 0.0509·34-s + ⋯
L(s)  = 1  + 0.184·2-s − 1.07·3-s − 0.966·4-s − 0.197·6-s − 0.481·7-s − 0.362·8-s + 0.152·9-s − 1.69·11-s + 1.03·12-s − 1.73·13-s − 0.0887·14-s + 0.899·16-s + 0.0473·17-s + 0.0280·18-s + 0.860·19-s + 0.517·21-s − 0.313·22-s + 1.68·23-s + 0.388·24-s − 0.319·26-s + 0.909·27-s + 0.465·28-s + 1.19·29-s − 1.85·31-s + 0.527·32-s + 1.82·33-s + 0.00873·34-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 \( 1 - 0.260T + 2T^{2} \)
3 \( 1 + 1.85T + 3T^{2} \)
7 \( 1 + 1.27T + 7T^{2} \)
11 \( 1 + 5.63T + 11T^{2} \)
13 \( 1 + 6.25T + 13T^{2} \)
17 \( 1 - 0.195T + 17T^{2} \)
19 \( 1 - 3.74T + 19T^{2} \)
23 \( 1 - 8.06T + 23T^{2} \)
29 \( 1 - 6.42T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 - 7.12T + 37T^{2} \)
41 \( 1 - 6.72T + 41T^{2} \)
43 \( 1 + 4.60T + 43T^{2} \)
47 \( 1 - 8.43T + 47T^{2} \)
53 \( 1 - 8.07T + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 + 5.12T + 61T^{2} \)
67 \( 1 + 6.67T + 67T^{2} \)
71 \( 1 + 4.71T + 71T^{2} \)
73 \( 1 + 9.61T + 73T^{2} \)
79 \( 1 + 5.87T + 79T^{2} \)
83 \( 1 - 3.59T + 83T^{2} \)
89 \( 1 - 0.879T + 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−7.45298655893568572311059481065, −7.19723103363509314727538842286, −5.98762852854251484486803529057, −5.37531033544724813328231001129, −5.04633615573645663692910705710, −4.40814918132580526086327317323, −3.09116395088129138138715270730, −2.62643827104319197952171209951, −0.801557244164014211387149945702, 0, 0.801557244164014211387149945702, 2.62643827104319197952171209951, 3.09116395088129138138715270730, 4.40814918132580526086327317323, 5.04633615573645663692910705710, 5.37531033544724813328231001129, 5.98762852854251484486803529057, 7.19723103363509314727538842286, 7.45298655893568572311059481065

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