Properties

Label 2-95e2-1.1-c1-0-137
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  − 1.48·2-s − 0.181·3-s + 0.193·4-s + 0.268·6-s + 1.30·7-s + 2.67·8-s − 2.96·9-s + 4.98·11-s − 0.0352·12-s − 0.406·13-s − 1.93·14-s − 4.35·16-s + 2.75·17-s + 4.39·18-s − 0.236·21-s − 7.38·22-s − 6.95·23-s − 0.485·24-s + 0.601·26-s + 1.08·27-s + 0.252·28-s + 4.01·29-s + 2.57·31-s + 1.09·32-s − 0.904·33-s − 4.07·34-s − 0.575·36-s + ⋯
L(s)  = 1  − 1.04·2-s − 0.104·3-s + 0.0969·4-s + 0.109·6-s + 0.492·7-s + 0.945·8-s − 0.989·9-s + 1.50·11-s − 0.0101·12-s − 0.112·13-s − 0.516·14-s − 1.08·16-s + 0.667·17-s + 1.03·18-s − 0.0516·21-s − 1.57·22-s − 1.45·23-s − 0.0991·24-s + 0.117·26-s + 0.208·27-s + 0.0478·28-s + 0.745·29-s + 0.462·31-s + 0.193·32-s − 0.157·33-s − 0.698·34-s − 0.0959·36-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

\(L(1)\) \(\approx\) \(1.039161044\)
\(L(\frac12)\) \(\approx\) \(1.039161044\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.48T + 2T^{2} \)
3 \( 1 + 0.181T + 3T^{2} \)
7 \( 1 - 1.30T + 7T^{2} \)
11 \( 1 - 4.98T + 11T^{2} \)
13 \( 1 + 0.406T + 13T^{2} \)
17 \( 1 - 2.75T + 17T^{2} \)
23 \( 1 + 6.95T + 23T^{2} \)
29 \( 1 - 4.01T + 29T^{2} \)
31 \( 1 - 2.57T + 31T^{2} \)
37 \( 1 - 3.71T + 37T^{2} \)
41 \( 1 - 1.21T + 41T^{2} \)
43 \( 1 + 3.12T + 43T^{2} \)
47 \( 1 - 6.50T + 47T^{2} \)
53 \( 1 - 6.32T + 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 0.934T + 61T^{2} \)
67 \( 1 - 5.29T + 67T^{2} \)
71 \( 1 - 1.63T + 71T^{2} \)
73 \( 1 + 7.68T + 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 - 15.2T + 83T^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 - 18.2T + 97T^{2} \)
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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

−8.163175408793345640357994278475, −7.23249162145242162092685809898, −6.42029638721044904298781241698, −5.85503816322934318066938145976, −4.92534453187641939205404289216, −4.24262796185785405853966366000, −3.48683056272208032368295251400, −2.34866796436169815428081275619, −1.45870595457070012641485834608, −0.63227148313379939320186834617, 0.63227148313379939320186834617, 1.45870595457070012641485834608, 2.34866796436169815428081275619, 3.48683056272208032368295251400, 4.24262796185785405853966366000, 4.92534453187641939205404289216, 5.85503816322934318066938145976, 6.42029638721044904298781241698, 7.23249162145242162092685809898, 8.163175408793345640357994278475

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