Properties

Label 2-2013-1.1-c3-0-25
Degree $2$
Conductor $2013$
Sign $1$
Analytic cond. $118.770$
Root an. cond. $10.8982$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.70·2-s − 3·3-s − 5.10·4-s + 17.1·5-s − 5.10·6-s − 24.4·7-s − 22.2·8-s + 9·9-s + 29.2·10-s − 11·11-s + 15.3·12-s − 72.7·13-s − 41.5·14-s − 51.5·15-s + 2.92·16-s − 61.9·17-s + 15.3·18-s − 26.2·19-s − 87.6·20-s + 73.2·21-s − 18.7·22-s − 177.·23-s + 66.8·24-s + 169.·25-s − 123.·26-s − 27·27-s + 124.·28-s + ⋯
L(s)  = 1  + 0.601·2-s − 0.577·3-s − 0.638·4-s + 1.53·5-s − 0.347·6-s − 1.31·7-s − 0.985·8-s + 0.333·9-s + 0.923·10-s − 0.301·11-s + 0.368·12-s − 1.55·13-s − 0.792·14-s − 0.886·15-s + 0.0457·16-s − 0.883·17-s + 0.200·18-s − 0.317·19-s − 0.980·20-s + 0.761·21-s − 0.181·22-s − 1.61·23-s + 0.568·24-s + 1.35·25-s − 0.932·26-s − 0.192·27-s + 0.841·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $1$
Analytic conductor: \(118.770\)
Root analytic conductor: \(10.8982\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2013,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8347824596\)
\(L(\frac12)\) \(\approx\) \(0.8347824596\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
11 \( 1 + 11T \)
61 \( 1 + 61T \)
good2 \( 1 - 1.70T + 8T^{2} \)
5 \( 1 - 17.1T + 125T^{2} \)
7 \( 1 + 24.4T + 343T^{2} \)
13 \( 1 + 72.7T + 2.19e3T^{2} \)
17 \( 1 + 61.9T + 4.91e3T^{2} \)
19 \( 1 + 26.2T + 6.85e3T^{2} \)
23 \( 1 + 177.T + 1.21e4T^{2} \)
29 \( 1 + 229.T + 2.43e4T^{2} \)
31 \( 1 - 143.T + 2.97e4T^{2} \)
37 \( 1 - 390.T + 5.06e4T^{2} \)
41 \( 1 + 23.0T + 6.89e4T^{2} \)
43 \( 1 - 128.T + 7.95e4T^{2} \)
47 \( 1 - 130.T + 1.03e5T^{2} \)
53 \( 1 + 362.T + 1.48e5T^{2} \)
59 \( 1 + 418.T + 2.05e5T^{2} \)
67 \( 1 - 360.T + 3.00e5T^{2} \)
71 \( 1 - 877.T + 3.57e5T^{2} \)
73 \( 1 - 313.T + 3.89e5T^{2} \)
79 \( 1 + 933.T + 4.93e5T^{2} \)
83 \( 1 - 87.3T + 5.71e5T^{2} \)
89 \( 1 - 106.T + 7.04e5T^{2} \)
97 \( 1 - 738.T + 9.12e5T^{2} \)
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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

−9.257744557111007217451566136298, −7.982464409675132583882309480114, −6.83011171033371265680568071811, −6.10905825190533102411819006707, −5.74304859297392630831292001263, −4.84738084258034340028230706438, −4.07464189701973621289202647950, −2.81131162609535556183275775289, −2.09952629073073132951031980882, −0.36981876021336517993632557321, 0.36981876021336517993632557321, 2.09952629073073132951031980882, 2.81131162609535556183275775289, 4.07464189701973621289202647950, 4.84738084258034340028230706438, 5.74304859297392630831292001263, 6.10905825190533102411819006707, 6.83011171033371265680568071811, 7.982464409675132583882309480114, 9.257744557111007217451566136298

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