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  + 2.36·2-s − 2.05·3-s + 3.57·4-s − 4.84·6-s − 3.07·7-s + 3.73·8-s + 1.21·9-s + 0.942·11-s − 7.34·12-s + 1.99·13-s − 7.26·14-s + 1.65·16-s + 3.11·17-s + 2.86·18-s − 6.33·19-s + 6.31·21-s + 2.22·22-s + 7.91·23-s − 7.65·24-s + 4.71·26-s + 3.66·27-s − 11.0·28-s − 1.63·29-s + 6.96·31-s − 3.55·32-s − 1.93·33-s + 7.35·34-s + ⋯
L(s)  = 1  + 1.67·2-s − 1.18·3-s + 1.78·4-s − 1.97·6-s − 1.16·7-s + 1.31·8-s + 0.404·9-s + 0.284·11-s − 2.12·12-s + 0.553·13-s − 1.94·14-s + 0.413·16-s + 0.755·17-s + 0.675·18-s − 1.45·19-s + 1.37·21-s + 0.474·22-s + 1.64·23-s − 1.56·24-s + 0.923·26-s + 0.705·27-s − 2.08·28-s − 0.303·29-s + 1.25·31-s − 0.628·32-s − 0.336·33-s + 1.26·34-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 \( 1 - 2.36T + 2T^{2} \)
3 \( 1 + 2.05T + 3T^{2} \)
7 \( 1 + 3.07T + 7T^{2} \)
11 \( 1 - 0.942T + 11T^{2} \)
13 \( 1 - 1.99T + 13T^{2} \)
17 \( 1 - 3.11T + 17T^{2} \)
19 \( 1 + 6.33T + 19T^{2} \)
23 \( 1 - 7.91T + 23T^{2} \)
29 \( 1 + 1.63T + 29T^{2} \)
31 \( 1 - 6.96T + 31T^{2} \)
37 \( 1 + 0.864T + 37T^{2} \)
41 \( 1 - 1.70T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + 6.49T + 47T^{2} \)
53 \( 1 - 2.20T + 53T^{2} \)
59 \( 1 + 7.69T + 59T^{2} \)
61 \( 1 + 3.70T + 61T^{2} \)
67 \( 1 + 7.96T + 67T^{2} \)
71 \( 1 + 7.02T + 71T^{2} \)
73 \( 1 + 6.13T + 73T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 - 3.98T + 83T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 + 17.6T + 97T^{2} \)
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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

−7.07855463606389040482597653129, −6.63015023574063899980557142140, −6.12870940230699106098934426535, −5.65115795932614064221189770982, −4.82010393346259508910302261741, −4.28191255653391638383710535429, −3.27215072478026113560341742239, −2.87335484482496593401610905664, −1.42132550743392716981273600246, 0, 1.42132550743392716981273600246, 2.87335484482496593401610905664, 3.27215072478026113560341742239, 4.28191255653391638383710535429, 4.82010393346259508910302261741, 5.65115795932614064221189770982, 6.12870940230699106098934426535, 6.63015023574063899980557142140, 7.07855463606389040482597653129

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