Properties

Label 2-95e2-1.1-c1-0-345
Degree $2$
Conductor $9025$
Sign $1$
Analytic cond. $72.0649$
Root an. cond. $8.48910$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.61·2-s − 2.23·3-s + 4.85·4-s − 5.85·6-s + 4.23·7-s + 7.47·8-s + 2.00·9-s + 5.47·11-s − 10.8·12-s − 2·13-s + 11.0·14-s + 9.85·16-s + 4.85·17-s + 5.23·18-s − 9.47·21-s + 14.3·22-s + 1.23·23-s − 16.7·24-s − 5.23·26-s + 2.23·27-s + 20.5·28-s − 6·29-s + 1.85·31-s + 10.8·32-s − 12.2·33-s + 12.7·34-s + 9.70·36-s + ⋯
L(s)  = 1  + 1.85·2-s − 1.29·3-s + 2.42·4-s − 2.38·6-s + 1.60·7-s + 2.64·8-s + 0.666·9-s + 1.64·11-s − 3.13·12-s − 0.554·13-s + 2.96·14-s + 2.46·16-s + 1.17·17-s + 1.23·18-s − 2.06·21-s + 3.05·22-s + 0.257·23-s − 3.41·24-s − 1.02·26-s + 0.430·27-s + 3.88·28-s − 1.11·29-s + 0.333·31-s + 1.91·32-s − 2.13·33-s + 2.17·34-s + 1.61·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9025\)    =    \(5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(72.0649\)
Root analytic conductor: \(8.48910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.658943430\)
\(L(\frac12)\) \(\approx\) \(6.658943430\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 - 2.61T + 2T^{2} \)
3 \( 1 + 2.23T + 3T^{2} \)
7 \( 1 - 4.23T + 7T^{2} \)
11 \( 1 - 5.47T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 4.85T + 17T^{2} \)
23 \( 1 - 1.23T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 1.85T + 31T^{2} \)
37 \( 1 - 4.14T + 37T^{2} \)
41 \( 1 - 6.38T + 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 - 7.61T + 47T^{2} \)
53 \( 1 + 6.38T + 53T^{2} \)
59 \( 1 - 2.76T + 59T^{2} \)
61 \( 1 + 5.56T + 61T^{2} \)
67 \( 1 + 8.70T + 67T^{2} \)
71 \( 1 - 4.52T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 + 1.09T + 83T^{2} \)
89 \( 1 + 3.70T + 89T^{2} \)
97 \( 1 - 7.38T + 97T^{2} \)
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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

−7.42544825868587483579585807805, −6.65800630512978560427129183879, −6.15172373135778251105009087537, −5.36523198740411731674930401879, −5.15533855330473720133972005760, −4.33576223688831134920011937676, −3.91745396200519847767147674306, −2.85071817820189945034282567084, −1.73361146595770377618383141504, −1.13781103842675887960957823458, 1.13781103842675887960957823458, 1.73361146595770377618383141504, 2.85071817820189945034282567084, 3.91745396200519847767147674306, 4.33576223688831134920011937676, 5.15533855330473720133972005760, 5.36523198740411731674930401879, 6.15172373135778251105009087537, 6.65800630512978560427129183879, 7.42544825868587483579585807805

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