Properties

Label 2-39e2-1.1-c1-0-45
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.56·2-s + 4.56·4-s + 0.561·5-s + 3.56·7-s + 6.56·8-s + 1.43·10-s − 2·11-s + 9.12·14-s + 7.68·16-s − 2.56·17-s + 1.12·19-s + 2.56·20-s − 5.12·22-s − 2·23-s − 4.68·25-s + 16.2·28-s + 5.68·29-s + 1.56·31-s + 6.56·32-s − 6.56·34-s + 2·35-s − 3.43·37-s + 2.87·38-s + 3.68·40-s + 2.56·41-s + 0.438·43-s − 9.12·44-s + ⋯
L(s)  = 1  + 1.81·2-s + 2.28·4-s + 0.251·5-s + 1.34·7-s + 2.31·8-s + 0.454·10-s − 0.603·11-s + 2.43·14-s + 1.92·16-s − 0.621·17-s + 0.257·19-s + 0.572·20-s − 1.09·22-s − 0.417·23-s − 0.936·25-s + 3.07·28-s + 1.05·29-s + 0.280·31-s + 1.15·32-s − 1.12·34-s + 0.338·35-s − 0.565·37-s + 0.466·38-s + 0.582·40-s + 0.400·41-s + 0.0668·43-s − 1.37·44-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: \(0\)
Selberg data: \((2,\ 1521,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.832556076\)
\(L(\frac12)\) \(\approx\) \(5.832556076\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.56T + 2T^{2} \)
5 \( 1 - 0.561T + 5T^{2} \)
7 \( 1 - 3.56T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
17 \( 1 + 2.56T + 17T^{2} \)
19 \( 1 - 1.12T + 19T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 - 5.68T + 29T^{2} \)
31 \( 1 - 1.56T + 31T^{2} \)
37 \( 1 + 3.43T + 37T^{2} \)
41 \( 1 - 2.56T + 41T^{2} \)
43 \( 1 - 0.438T + 43T^{2} \)
47 \( 1 + 8.24T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 + 0.438T + 67T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 - 1.87T + 73T^{2} \)
79 \( 1 - 9.56T + 79T^{2} \)
83 \( 1 + 9.12T + 83T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 - 4.43T + 97T^{2} \)
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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

−9.657833910237604544020920023976, −8.300635451467443019539945326496, −7.73931305784082476419620846140, −6.71767869519000070985174829874, −5.96063859230283993864858538008, −5.04062066361909862678800800901, −4.67577561131754230408858777349, −3.65157862846727552304585172851, −2.52774417137176119618141532376, −1.70604796692066655437177248167, 1.70604796692066655437177248167, 2.52774417137176119618141532376, 3.65157862846727552304585172851, 4.67577561131754230408858777349, 5.04062066361909862678800800901, 5.96063859230283993864858538008, 6.71767869519000070985174829874, 7.73931305784082476419620846140, 8.300635451467443019539945326496, 9.657833910237604544020920023976

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