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  + 1.49·2-s − 3.12·3-s + 0.226·4-s − 4.66·6-s − 4.66·7-s − 2.64·8-s + 6.77·9-s − 5.92·11-s − 0.707·12-s + 4.29·13-s − 6.95·14-s − 4.40·16-s + 5.72·17-s + 10.1·18-s − 1.23·19-s + 14.5·21-s − 8.83·22-s + 3.63·23-s + 8.27·24-s + 6.40·26-s − 11.8·27-s − 1.05·28-s + 1.81·29-s − 1.11·31-s − 1.27·32-s + 18.5·33-s + 8.54·34-s + ⋯
L(s)  = 1  + 1.05·2-s − 1.80·3-s + 0.113·4-s − 1.90·6-s − 1.76·7-s − 0.935·8-s + 2.25·9-s − 1.78·11-s − 0.204·12-s + 1.19·13-s − 1.85·14-s − 1.10·16-s + 1.38·17-s + 2.38·18-s − 0.282·19-s + 3.17·21-s − 1.88·22-s + 0.757·23-s + 1.68·24-s + 1.25·26-s − 2.27·27-s − 0.199·28-s + 0.336·29-s − 0.200·31-s − 0.225·32-s + 3.22·33-s + 1.46·34-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 \( 1 - 1.49T + 2T^{2} \)
3 \( 1 + 3.12T + 3T^{2} \)
7 \( 1 + 4.66T + 7T^{2} \)
11 \( 1 + 5.92T + 11T^{2} \)
13 \( 1 - 4.29T + 13T^{2} \)
17 \( 1 - 5.72T + 17T^{2} \)
19 \( 1 + 1.23T + 19T^{2} \)
23 \( 1 - 3.63T + 23T^{2} \)
29 \( 1 - 1.81T + 29T^{2} \)
31 \( 1 + 1.11T + 31T^{2} \)
37 \( 1 - 7.32T + 37T^{2} \)
41 \( 1 + 6.45T + 41T^{2} \)
43 \( 1 - 4.57T + 43T^{2} \)
47 \( 1 - 7.71T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + 5.66T + 59T^{2} \)
61 \( 1 - 4.76T + 61T^{2} \)
67 \( 1 - 14.0T + 67T^{2} \)
71 \( 1 + 3.96T + 71T^{2} \)
73 \( 1 + 8.92T + 73T^{2} \)
79 \( 1 - 1.77T + 79T^{2} \)
83 \( 1 - 9.93T + 83T^{2} \)
89 \( 1 + 7.03T + 89T^{2} \)
97 \( 1 + 5.03T + 97T^{2} \)
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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

−7.33523510978742456620593929238, −6.57003963519161240550902174003, −6.03723409839425120188198049386, −5.59738776202048974656370667532, −5.09401097983024821674419923129, −4.19739490533648980409932389797, −3.41243609815339232984225478877, −2.73847108413064648621492668275, −0.896046559961821367083646317806, 0, 0.896046559961821367083646317806, 2.73847108413064648621492668275, 3.41243609815339232984225478877, 4.19739490533648980409932389797, 5.09401097983024821674419923129, 5.59738776202048974656370667532, 6.03723409839425120188198049386, 6.57003963519161240550902174003, 7.33523510978742456620593929238

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