Properties

Label 2-39e2-1.1-c1-0-43
Degree $2$
Conductor $1521$
Sign $-1$
Analytic cond. $12.1452$
Root an. cond. $3.48500$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.445·2-s − 1.80·4-s − 0.246·5-s − 1.75·7-s − 1.69·8-s − 0.109·10-s + 5.65·11-s − 0.780·14-s + 2.85·16-s + 3.80·17-s − 5.58·19-s + 0.445·20-s + 2.51·22-s − 8.34·23-s − 4.93·25-s + 3.15·28-s + 5.93·29-s − 5.26·31-s + 4.65·32-s + 1.69·34-s + 0.432·35-s − 3.19·37-s − 2.48·38-s + 0.417·40-s + 0.445·41-s + 1.71·43-s − 10.1·44-s + ⋯
L(s)  = 1  + 0.314·2-s − 0.900·4-s − 0.110·5-s − 0.662·7-s − 0.598·8-s − 0.0347·10-s + 1.70·11-s − 0.208·14-s + 0.712·16-s + 0.922·17-s − 1.28·19-s + 0.0995·20-s + 0.536·22-s − 1.74·23-s − 0.987·25-s + 0.596·28-s + 1.10·29-s − 0.946·31-s + 0.822·32-s + 0.290·34-s + 0.0731·35-s − 0.525·37-s − 0.403·38-s + 0.0660·40-s + 0.0695·41-s + 0.261·43-s − 1.53·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(12.1452\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1521,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 - 0.445T + 2T^{2} \)
5 \( 1 + 0.246T + 5T^{2} \)
7 \( 1 + 1.75T + 7T^{2} \)
11 \( 1 - 5.65T + 11T^{2} \)
17 \( 1 - 3.80T + 17T^{2} \)
19 \( 1 + 5.58T + 19T^{2} \)
23 \( 1 + 8.34T + 23T^{2} \)
29 \( 1 - 5.93T + 29T^{2} \)
31 \( 1 + 5.26T + 31T^{2} \)
37 \( 1 + 3.19T + 37T^{2} \)
41 \( 1 - 0.445T + 41T^{2} \)
43 \( 1 - 1.71T + 43T^{2} \)
47 \( 1 + 6.73T + 47T^{2} \)
53 \( 1 - 1.06T + 53T^{2} \)
59 \( 1 + 13.7T + 59T^{2} \)
61 \( 1 + 8.51T + 61T^{2} \)
67 \( 1 + 5.96T + 67T^{2} \)
71 \( 1 + 5.71T + 71T^{2} \)
73 \( 1 + 7.35T + 73T^{2} \)
79 \( 1 - 4.45T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 0.137T + 89T^{2} \)
97 \( 1 - 13.6T + 97T^{2} \)
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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

−9.097897387205477670037995900352, −8.423757190499622701131216757011, −7.51418938028855006725284722294, −6.18428676484567164978347077363, −6.09284928634346219876523376365, −4.64982782161842097768221998969, −3.95115751257984579506791085647, −3.29718836709218914119442886516, −1.63451042249254106473523472232, 0, 1.63451042249254106473523472232, 3.29718836709218914119442886516, 3.95115751257984579506791085647, 4.64982782161842097768221998969, 6.09284928634346219876523376365, 6.18428676484567164978347077363, 7.51418938028855006725284722294, 8.423757190499622701131216757011, 9.097897387205477670037995900352

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