Properties

Label 2-39e2-1.1-c1-0-0
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  − 2.23·2-s + 3.00·4-s − 2.23·5-s − 3.46·7-s − 2.23·8-s + 5.00·10-s − 4.47·11-s + 7.74·14-s − 0.999·16-s − 3.87·17-s + 3.46·19-s − 6.70·20-s + 10.0·22-s − 7.74·23-s − 10.3·28-s − 3.87·29-s + 6.70·32-s + 8.66·34-s + 7.74·35-s − 1.73·37-s − 7.74·38-s + 5.00·40-s − 2.23·41-s − 2·43-s − 13.4·44-s + 17.3·46-s − 4.47·47-s + ⋯
L(s)  = 1  − 1.58·2-s + 1.50·4-s − 0.999·5-s − 1.30·7-s − 0.790·8-s + 1.58·10-s − 1.34·11-s + 2.07·14-s − 0.249·16-s − 0.939·17-s + 0.794·19-s − 1.49·20-s + 2.13·22-s − 1.61·23-s − 1.96·28-s − 0.719·29-s + 1.18·32-s + 1.48·34-s + 1.30·35-s − 0.284·37-s − 1.25·38-s + 0.790·40-s − 0.349·41-s − 0.304·43-s − 2.02·44-s + 2.55·46-s − 0.652·47-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1286954548\)
\(L(\frac12)\) \(\approx\) \(0.1286954548\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23T + 2T^{2} \)
5 \( 1 + 2.23T + 5T^{2} \)
7 \( 1 + 3.46T + 7T^{2} \)
11 \( 1 + 4.47T + 11T^{2} \)
17 \( 1 + 3.87T + 17T^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 + 7.74T + 23T^{2} \)
29 \( 1 + 3.87T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 1.73T + 37T^{2} \)
41 \( 1 + 2.23T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 4.47T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 - 8.94T + 59T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 + 3.46T + 67T^{2} \)
71 \( 1 - 4.47T + 71T^{2} \)
73 \( 1 + 15.5T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 4.47T + 83T^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 + 6.92T + 97T^{2} \)
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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

−9.516116731833539495334961519875, −8.626960576478567310644938781275, −7.958481422432296433980684379996, −7.36757050196155756063856695066, −6.63339800447605068874570904626, −5.58706080568863421411953848254, −4.22225059875039158251176681207, −3.18262028694788129275737234824, −2.12319073653516606276439508814, −0.29872830869381274131204375417, 0.29872830869381274131204375417, 2.12319073653516606276439508814, 3.18262028694788129275737234824, 4.22225059875039158251176681207, 5.58706080568863421411953848254, 6.63339800447605068874570904626, 7.36757050196155756063856695066, 7.958481422432296433980684379996, 8.626960576478567310644938781275, 9.516116731833539495334961519875

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