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  − 0.257·2-s − 3.28·3-s − 1.93·4-s + 0.843·6-s + 3.60·7-s + 1.01·8-s + 7.77·9-s − 1.43·11-s + 6.34·12-s + 2.88·13-s − 0.926·14-s + 3.60·16-s + 0.875·17-s − 1.99·18-s + 2.64·19-s − 11.8·21-s + 0.369·22-s − 4.70·23-s − 3.31·24-s − 0.741·26-s − 15.6·27-s − 6.97·28-s − 3.36·29-s − 0.807·31-s − 2.95·32-s + 4.72·33-s − 0.225·34-s + ⋯
L(s)  = 1  − 0.181·2-s − 1.89·3-s − 0.966·4-s + 0.344·6-s + 1.36·7-s + 0.357·8-s + 2.59·9-s − 0.433·11-s + 1.83·12-s + 0.799·13-s − 0.247·14-s + 0.901·16-s + 0.212·17-s − 0.470·18-s + 0.606·19-s − 2.58·21-s + 0.0788·22-s − 0.980·23-s − 0.677·24-s − 0.145·26-s − 3.01·27-s − 1.31·28-s − 0.624·29-s − 0.145·31-s − 0.521·32-s + 0.822·33-s − 0.0386·34-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 \( 1 + 0.257T + 2T^{2} \)
3 \( 1 + 3.28T + 3T^{2} \)
7 \( 1 - 3.60T + 7T^{2} \)
11 \( 1 + 1.43T + 11T^{2} \)
13 \( 1 - 2.88T + 13T^{2} \)
17 \( 1 - 0.875T + 17T^{2} \)
19 \( 1 - 2.64T + 19T^{2} \)
23 \( 1 + 4.70T + 23T^{2} \)
29 \( 1 + 3.36T + 29T^{2} \)
31 \( 1 + 0.807T + 31T^{2} \)
37 \( 1 - 3.55T + 37T^{2} \)
41 \( 1 + 9.22T + 41T^{2} \)
43 \( 1 + 4.15T + 43T^{2} \)
47 \( 1 - 1.63T + 47T^{2} \)
53 \( 1 - 7.56T + 53T^{2} \)
59 \( 1 + 14.0T + 59T^{2} \)
61 \( 1 - 8.82T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + 8.00T + 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 + 0.971T + 83T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 - 3.33T + 97T^{2} \)
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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

−7.72984559915791712710207260836, −7.02942160324812105673729057581, −5.95280024907213385701753237690, −5.56545161829091838590410677281, −4.92036962147061894253252133937, −4.40550467822431416659296730629, −3.62366356208751776136915084261, −1.75365513431516678249786302452, −1.08681445983356670860840198757, 0, 1.08681445983356670860840198757, 1.75365513431516678249786302452, 3.62366356208751776136915084261, 4.40550467822431416659296730629, 4.92036962147061894253252133937, 5.56545161829091838590410677281, 5.95280024907213385701753237690, 7.02942160324812105673729057581, 7.72984559915791712710207260836

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