Properties

Label 2-95e2-1.1-c1-0-184
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  + 2.23·2-s − 2·3-s + 3.00·4-s − 4.47·6-s − 1.23·7-s + 2.23·8-s + 9-s + 5.23·11-s − 6.00·12-s + 1.85·13-s − 2.76·14-s − 0.999·16-s − 0.618·17-s + 2.23·18-s + 2.47·21-s + 11.7·22-s + 4.47·23-s − 4.47·24-s + 4.14·26-s + 4·27-s − 3.70·28-s + 7.09·29-s − 6·31-s − 6.70·32-s − 10.4·33-s − 1.38·34-s + 3.00·36-s + ⋯
L(s)  = 1  + 1.58·2-s − 1.15·3-s + 1.50·4-s − 1.82·6-s − 0.467·7-s + 0.790·8-s + 0.333·9-s + 1.57·11-s − 1.73·12-s + 0.514·13-s − 0.738·14-s − 0.249·16-s − 0.149·17-s + 0.527·18-s + 0.539·21-s + 2.49·22-s + 0.932·23-s − 0.912·24-s + 0.813·26-s + 0.769·27-s − 0.700·28-s + 1.31·29-s − 1.07·31-s − 1.18·32-s − 1.82·33-s − 0.237·34-s + 0.500·36-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

\(L(1)\) \(\approx\) \(3.413918733\)
\(L(\frac12)\) \(\approx\) \(3.413918733\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23T + 2T^{2} \)
3 \( 1 + 2T + 3T^{2} \)
7 \( 1 + 1.23T + 7T^{2} \)
11 \( 1 - 5.23T + 11T^{2} \)
13 \( 1 - 1.85T + 13T^{2} \)
17 \( 1 + 0.618T + 17T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 - 7.09T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 8.85T + 37T^{2} \)
41 \( 1 - 4.09T + 41T^{2} \)
43 \( 1 - 0.472T + 43T^{2} \)
47 \( 1 + 4.76T + 47T^{2} \)
53 \( 1 - 0.0901T + 53T^{2} \)
59 \( 1 - 3.23T + 59T^{2} \)
61 \( 1 - 9.85T + 61T^{2} \)
67 \( 1 - 0.472T + 67T^{2} \)
71 \( 1 + 3.23T + 71T^{2} \)
73 \( 1 - 14.5T + 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 - 9.70T + 83T^{2} \)
89 \( 1 + 7.32T + 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−7.09367141551672118955297101100, −6.63464541691598933812168329144, −6.33470489218713852908434629411, −5.56820953714516473098882604382, −5.04516660089517054094465786514, −4.33269845928362546909768716229, −3.62532803910219176447882149536, −3.05634352567261780926335228498, −1.84740275996839805231171468919, −0.75153563844865592177481562054, 0.75153563844865592177481562054, 1.84740275996839805231171468919, 3.05634352567261780926335228498, 3.62532803910219176447882149536, 4.33269845928362546909768716229, 5.04516660089517054094465786514, 5.56820953714516473098882604382, 6.33470489218713852908434629411, 6.63464541691598933812168329144, 7.09367141551672118955297101100

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