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  − 0.487·2-s + 0.815·3-s − 1.76·4-s − 0.397·6-s + 4.61·7-s + 1.83·8-s − 2.33·9-s + 1.93·11-s − 1.43·12-s + 3.85·13-s − 2.24·14-s + 2.63·16-s − 5.40·17-s + 1.13·18-s − 4.17·19-s + 3.76·21-s − 0.945·22-s + 1.42·23-s + 1.49·24-s − 1.87·26-s − 4.34·27-s − 8.13·28-s − 4.85·29-s − 7.24·31-s − 4.94·32-s + 1.58·33-s + 2.63·34-s + ⋯
L(s)  = 1  − 0.344·2-s + 0.470·3-s − 0.881·4-s − 0.162·6-s + 1.74·7-s + 0.648·8-s − 0.778·9-s + 0.584·11-s − 0.414·12-s + 1.06·13-s − 0.600·14-s + 0.657·16-s − 1.31·17-s + 0.268·18-s − 0.956·19-s + 0.820·21-s − 0.201·22-s + 0.297·23-s + 0.305·24-s − 0.368·26-s − 0.836·27-s − 1.53·28-s − 0.902·29-s − 1.30·31-s − 0.874·32-s + 0.275·33-s + 0.451·34-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 \( 1 + 0.487T + 2T^{2} \)
3 \( 1 - 0.815T + 3T^{2} \)
7 \( 1 - 4.61T + 7T^{2} \)
11 \( 1 - 1.93T + 11T^{2} \)
13 \( 1 - 3.85T + 13T^{2} \)
17 \( 1 + 5.40T + 17T^{2} \)
19 \( 1 + 4.17T + 19T^{2} \)
23 \( 1 - 1.42T + 23T^{2} \)
29 \( 1 + 4.85T + 29T^{2} \)
31 \( 1 + 7.24T + 31T^{2} \)
37 \( 1 + 7.12T + 37T^{2} \)
41 \( 1 - 9.18T + 41T^{2} \)
43 \( 1 + 2.93T + 43T^{2} \)
47 \( 1 + 2.48T + 47T^{2} \)
53 \( 1 + 5.64T + 53T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 + 13.9T + 61T^{2} \)
67 \( 1 - 7.30T + 67T^{2} \)
71 \( 1 + 14.8T + 71T^{2} \)
73 \( 1 + 0.240T + 73T^{2} \)
79 \( 1 + 3.15T + 79T^{2} \)
83 \( 1 - 2.46T + 83T^{2} \)
89 \( 1 - 5.55T + 89T^{2} \)
97 \( 1 + 5.27T + 97T^{2} \)
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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

−7.957959236040833281400661875979, −7.33706254099356660782277267816, −6.21594149198485462916819670921, −5.51614252384746702724530538302, −4.66060195125408808224910144222, −4.17324502459963146715233220750, −3.35358439270987767988950283187, −2.00082004079458772946180460546, −1.46950034316185088932094245061, 0, 1.46950034316185088932094245061, 2.00082004079458772946180460546, 3.35358439270987767988950283187, 4.17324502459963146715233220750, 4.66060195125408808224910144222, 5.51614252384746702724530538302, 6.21594149198485462916819670921, 7.33706254099356660782277267816, 7.957959236040833281400661875979

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