Properties

Label 2-95e2-1.1-c1-0-26
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  − 2·4-s − 4.35·7-s − 3·9-s + 5·11-s + 4·16-s − 4.35·17-s − 8.71·23-s + 8.71·28-s + 6·36-s − 13.0·43-s − 10·44-s − 4.35·47-s + 12.0·49-s − 15·61-s + 13.0·63-s − 8·64-s + 8.71·68-s − 13.0·73-s − 21.7·77-s + 9·81-s + 8.71·83-s + 17.4·92-s − 15·99-s − 10·101-s − 17.4·112-s + 19.0·119-s + ⋯
L(s)  = 1  − 4-s − 1.64·7-s − 9-s + 1.50·11-s + 16-s − 1.05·17-s − 1.81·23-s + 1.64·28-s + 36-s − 1.99·43-s − 1.50·44-s − 0.635·47-s + 1.71·49-s − 1.92·61-s + 1.64·63-s − 64-s + 1.05·68-s − 1.53·73-s − 2.48·77-s + 81-s + 0.956·83-s + 1.81·92-s − 1.50·99-s − 0.995·101-s − 1.64·112-s + 1.74·119-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

\(L(1)\) \(\approx\) \(0.3054758285\)
\(L(\frac12)\) \(\approx\) \(0.3054758285\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T^{2} \)
3 \( 1 + 3T^{2} \)
7 \( 1 + 4.35T + 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 4.35T + 17T^{2} \)
23 \( 1 + 8.71T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 13.0T + 43T^{2} \)
47 \( 1 + 4.35T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 15T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 8.71T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−7.87228133581023497482038977982, −6.82889104615404670396933503724, −6.24537487885028307234805971327, −5.93752055122709237833610759312, −4.88929259409178355287874676243, −4.05796251768825214799445401206, −3.57905045340492475134955966368, −2.86826313090269554247123305551, −1.66489463975726893291534958192, −0.26297038550728600901815315916, 0.26297038550728600901815315916, 1.66489463975726893291534958192, 2.86826313090269554247123305551, 3.57905045340492475134955966368, 4.05796251768825214799445401206, 4.88929259409178355287874676243, 5.93752055122709237833610759312, 6.24537487885028307234805971327, 6.82889104615404670396933503724, 7.87228133581023497482038977982

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