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  − 1.70·2-s − 1.38·3-s + 0.922·4-s + 2.36·6-s − 5.05·7-s + 1.84·8-s − 1.08·9-s − 5.25·11-s − 1.27·12-s + 2.84·13-s + 8.64·14-s − 4.99·16-s + 6.16·17-s + 1.85·18-s + 1.47·19-s + 6.99·21-s + 8.97·22-s − 2.54·23-s − 2.54·24-s − 4.86·26-s + 5.65·27-s − 4.66·28-s − 9.53·29-s − 7.94·31-s + 4.85·32-s + 7.26·33-s − 10.5·34-s + ⋯
L(s)  = 1  − 1.20·2-s − 0.798·3-s + 0.461·4-s + 0.965·6-s − 1.91·7-s + 0.651·8-s − 0.362·9-s − 1.58·11-s − 0.368·12-s + 0.788·13-s + 2.31·14-s − 1.24·16-s + 1.49·17-s + 0.437·18-s + 0.337·19-s + 1.52·21-s + 1.91·22-s − 0.530·23-s − 0.520·24-s − 0.953·26-s + 1.08·27-s − 0.882·28-s − 1.77·29-s − 1.42·31-s + 0.858·32-s + 1.26·33-s − 1.80·34-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 \( 1 + 1.70T + 2T^{2} \)
3 \( 1 + 1.38T + 3T^{2} \)
7 \( 1 + 5.05T + 7T^{2} \)
11 \( 1 + 5.25T + 11T^{2} \)
13 \( 1 - 2.84T + 13T^{2} \)
17 \( 1 - 6.16T + 17T^{2} \)
19 \( 1 - 1.47T + 19T^{2} \)
23 \( 1 + 2.54T + 23T^{2} \)
29 \( 1 + 9.53T + 29T^{2} \)
31 \( 1 + 7.94T + 31T^{2} \)
37 \( 1 + 8.85T + 37T^{2} \)
41 \( 1 - 1.60T + 41T^{2} \)
43 \( 1 + 0.0179T + 43T^{2} \)
47 \( 1 - 5.95T + 47T^{2} \)
53 \( 1 - 8.07T + 53T^{2} \)
59 \( 1 + 3.28T + 59T^{2} \)
61 \( 1 - 7.19T + 61T^{2} \)
67 \( 1 - 2.92T + 67T^{2} \)
71 \( 1 + 4.49T + 71T^{2} \)
73 \( 1 + 4.29T + 73T^{2} \)
79 \( 1 - 2.34T + 79T^{2} \)
83 \( 1 - 9.27T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + 11.0T + 97T^{2} \)
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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

−7.53076196277875454874208683801, −7.36547651377551945213529323272, −6.30345259395773368592797910494, −5.61196210541562468577248152601, −5.29437534012843565419245031654, −3.77571676894305470932770492300, −3.22756919990619051552848851092, −2.12530464167535625134538013437, −0.71034645181855999942059461395, 0, 0.71034645181855999942059461395, 2.12530464167535625134538013437, 3.22756919990619051552848851092, 3.77571676894305470932770492300, 5.29437534012843565419245031654, 5.61196210541562468577248152601, 6.30345259395773368592797910494, 7.36547651377551945213529323272, 7.53076196277875454874208683801

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