Properties

Label 2-95e2-1.1-c1-0-231
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  + 1.22·2-s + 2.28·3-s − 0.507·4-s + 2.79·6-s + 1.28·7-s − 3.06·8-s + 2.22·9-s + 0.285·11-s − 1.15·12-s − 5·13-s + 1.57·14-s − 2.72·16-s + 6.23·17-s + 2.71·18-s + 2.93·21-s + 0.348·22-s + 5.23·23-s − 7.00·24-s − 6.10·26-s − 1.77·27-s − 0.651·28-s − 1.28·29-s + 1.22·31-s + 2.79·32-s + 0.651·33-s + 7.61·34-s − 1.12·36-s + ⋯
L(s)  = 1  + 0.863·2-s + 1.31·3-s − 0.253·4-s + 1.13·6-s + 0.485·7-s − 1.08·8-s + 0.740·9-s + 0.0859·11-s − 0.334·12-s − 1.38·13-s + 0.419·14-s − 0.682·16-s + 1.51·17-s + 0.639·18-s + 0.640·21-s + 0.0742·22-s + 1.09·23-s − 1.42·24-s − 1.19·26-s − 0.342·27-s − 0.123·28-s − 0.238·29-s + 0.219·31-s + 0.493·32-s + 0.113·33-s + 1.30·34-s − 0.187·36-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

\(L(1)\) \(\approx\) \(4.651008126\)
\(L(\frac12)\) \(\approx\) \(4.651008126\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.22T + 2T^{2} \)
3 \( 1 - 2.28T + 3T^{2} \)
7 \( 1 - 1.28T + 7T^{2} \)
11 \( 1 - 0.285T + 11T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 - 6.23T + 17T^{2} \)
23 \( 1 - 5.23T + 23T^{2} \)
29 \( 1 + 1.28T + 29T^{2} \)
31 \( 1 - 1.22T + 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 - 0.841T + 41T^{2} \)
43 \( 1 - 4.95T + 43T^{2} \)
47 \( 1 + 5.72T + 47T^{2} \)
53 \( 1 - 12.3T + 53T^{2} \)
59 \( 1 + 5.72T + 59T^{2} \)
61 \( 1 - 4.45T + 61T^{2} \)
67 \( 1 + 0.985T + 67T^{2} \)
71 \( 1 - 2.92T + 71T^{2} \)
73 \( 1 - 0.764T + 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + 1.66T + 83T^{2} \)
89 \( 1 - 16.0T + 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−7.79058049445732013317371420272, −7.25925557931517146436889409193, −6.24930388399633763035815927256, −5.40338298601629609071790711714, −4.89376164278347901404634319820, −4.17304330550342576888938789203, −3.40198203981023653644326089741, −2.84330891131640788476667724823, −2.17087887967364315871950223725, −0.859670514604752430783571246779, 0.859670514604752430783571246779, 2.17087887967364315871950223725, 2.84330891131640788476667724823, 3.40198203981023653644326089741, 4.17304330550342576888938789203, 4.89376164278347901404634319820, 5.40338298601629609071790711714, 6.24930388399633763035815927256, 7.25925557931517146436889409193, 7.79058049445732013317371420272

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