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  − 0.636·2-s + 2.24·3-s − 1.59·4-s − 1.43·6-s + 1.14·7-s + 2.28·8-s + 2.05·9-s − 0.440·11-s − 3.58·12-s + 0.0742·13-s − 0.729·14-s + 1.73·16-s − 3.83·17-s − 1.30·18-s − 4.81·19-s + 2.57·21-s + 0.280·22-s + 6.24·23-s + 5.14·24-s − 0.0472·26-s − 2.13·27-s − 1.82·28-s + 2.54·29-s − 0.429·31-s − 5.68·32-s − 0.989·33-s + 2.44·34-s + ⋯
L(s)  = 1  − 0.450·2-s + 1.29·3-s − 0.797·4-s − 0.584·6-s + 0.432·7-s + 0.809·8-s + 0.683·9-s − 0.132·11-s − 1.03·12-s + 0.0205·13-s − 0.194·14-s + 0.432·16-s − 0.930·17-s − 0.307·18-s − 1.10·19-s + 0.561·21-s + 0.0597·22-s + 1.30·23-s + 1.04·24-s − 0.00927·26-s − 0.410·27-s − 0.345·28-s + 0.472·29-s − 0.0771·31-s − 1.00·32-s − 0.172·33-s + 0.418·34-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 \( 1 + 0.636T + 2T^{2} \)
3 \( 1 - 2.24T + 3T^{2} \)
7 \( 1 - 1.14T + 7T^{2} \)
11 \( 1 + 0.440T + 11T^{2} \)
13 \( 1 - 0.0742T + 13T^{2} \)
17 \( 1 + 3.83T + 17T^{2} \)
19 \( 1 + 4.81T + 19T^{2} \)
23 \( 1 - 6.24T + 23T^{2} \)
29 \( 1 - 2.54T + 29T^{2} \)
31 \( 1 + 0.429T + 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 + 4.91T + 41T^{2} \)
43 \( 1 + 1.72T + 43T^{2} \)
47 \( 1 - 0.485T + 47T^{2} \)
53 \( 1 + 2.29T + 53T^{2} \)
59 \( 1 + 1.98T + 59T^{2} \)
61 \( 1 - 8.96T + 61T^{2} \)
67 \( 1 - 1.53T + 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 - 1.84T + 79T^{2} \)
83 \( 1 + 1.08T + 83T^{2} \)
89 \( 1 - 1.27T + 89T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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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

−8.116369674493026406215616652906, −7.21515936722205611864721619553, −6.60059394338653759195654569785, −5.33473464271359478220308527761, −4.72607325670691638254110673725, −3.96667065410571380527540120249, −3.21868376397996857095144802704, −2.26296070916565445152462898207, −1.45460244846202286139555833701, 0, 1.45460244846202286139555833701, 2.26296070916565445152462898207, 3.21868376397996857095144802704, 3.96667065410571380527540120249, 4.72607325670691638254110673725, 5.33473464271359478220308527761, 6.60059394338653759195654569785, 7.21515936722205611864721619553, 8.116369674493026406215616652906

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