Properties

Label 2-4923-1.1-c1-0-73
Degree $2$
Conductor $4923$
Sign $1$
Analytic cond. $39.3103$
Root an. cond. $6.26979$
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  + 0.763·2-s − 1.41·4-s + 1.51·5-s + 1.20·7-s − 2.60·8-s + 1.15·10-s + 5.83·11-s − 5.40·13-s + 0.917·14-s + 0.842·16-s − 1.92·17-s + 0.965·19-s − 2.14·20-s + 4.45·22-s + 0.470·23-s − 2.71·25-s − 4.12·26-s − 1.70·28-s + 3.67·29-s + 0.675·31-s + 5.86·32-s − 1.47·34-s + 1.81·35-s + 5.66·37-s + 0.737·38-s − 3.94·40-s + 5.00·41-s + ⋯
L(s)  = 1  + 0.539·2-s − 0.708·4-s + 0.676·5-s + 0.454·7-s − 0.922·8-s + 0.365·10-s + 1.76·11-s − 1.49·13-s + 0.245·14-s + 0.210·16-s − 0.467·17-s + 0.221·19-s − 0.479·20-s + 0.950·22-s + 0.0980·23-s − 0.542·25-s − 0.809·26-s − 0.321·28-s + 0.682·29-s + 0.121·31-s + 1.03·32-s − 0.252·34-s + 0.307·35-s + 0.932·37-s + 0.119·38-s − 0.623·40-s + 0.781·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4923\)    =    \(3^{2} \cdot 547\)
Sign: $1$
Analytic conductor: \(39.3103\)
Root analytic conductor: \(6.26979\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4923,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.506395926\)
\(L(\frac12)\) \(\approx\) \(2.506395926\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
547 \( 1 + T \)
good2 \( 1 - 0.763T + 2T^{2} \)
5 \( 1 - 1.51T + 5T^{2} \)
7 \( 1 - 1.20T + 7T^{2} \)
11 \( 1 - 5.83T + 11T^{2} \)
13 \( 1 + 5.40T + 13T^{2} \)
17 \( 1 + 1.92T + 17T^{2} \)
19 \( 1 - 0.965T + 19T^{2} \)
23 \( 1 - 0.470T + 23T^{2} \)
29 \( 1 - 3.67T + 29T^{2} \)
31 \( 1 - 0.675T + 31T^{2} \)
37 \( 1 - 5.66T + 37T^{2} \)
41 \( 1 - 5.00T + 41T^{2} \)
43 \( 1 - 3.18T + 43T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 + 8.25T + 53T^{2} \)
59 \( 1 - 5.23T + 59T^{2} \)
61 \( 1 + 3.53T + 61T^{2} \)
67 \( 1 - 8.70T + 67T^{2} \)
71 \( 1 - 9.50T + 71T^{2} \)
73 \( 1 - 9.68T + 73T^{2} \)
79 \( 1 + 5.23T + 79T^{2} \)
83 \( 1 + 14.9T + 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + 6.07T + 97T^{2} \)
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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

−8.357382604473647630301234457201, −7.48155049789642153704238257977, −6.63399080049048500829537409502, −6.00222075078023995372167806290, −5.24968983588362512235615045704, −4.48105332458942430670542600070, −4.04042595911871352435567866005, −2.91521649974526160893034823874, −1.99457949400417906883960970589, −0.822741011537124108930691896517, 0.822741011537124108930691896517, 1.99457949400417906883960970589, 2.91521649974526160893034823874, 4.04042595911871352435567866005, 4.48105332458942430670542600070, 5.24968983588362512235615045704, 6.00222075078023995372167806290, 6.63399080049048500829537409502, 7.48155049789642153704238257977, 8.357382604473647630301234457201

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