Properties

Label 2-2013-1.1-c3-0-124
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  + 0.690·2-s − 3·3-s − 7.52·4-s + 4.55·5-s − 2.07·6-s + 26.5·7-s − 10.7·8-s + 9·9-s + 3.14·10-s + 11·11-s + 22.5·12-s + 14.5·13-s + 18.2·14-s − 13.6·15-s + 52.7·16-s + 68.2·17-s + 6.21·18-s + 139.·19-s − 34.2·20-s − 79.5·21-s + 7.59·22-s + 17.3·23-s + 32.1·24-s − 104.·25-s + 10.0·26-s − 27·27-s − 199.·28-s + ⋯
L(s)  = 1  + 0.243·2-s − 0.577·3-s − 0.940·4-s + 0.407·5-s − 0.140·6-s + 1.43·7-s − 0.473·8-s + 0.333·9-s + 0.0994·10-s + 0.301·11-s + 0.542·12-s + 0.310·13-s + 0.349·14-s − 0.235·15-s + 0.824·16-s + 0.973·17-s + 0.0813·18-s + 1.68·19-s − 0.383·20-s − 0.826·21-s + 0.0735·22-s + 0.157·23-s + 0.273·24-s − 0.833·25-s + 0.0757·26-s − 0.192·27-s − 1.34·28-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

\(L(2)\) \(\approx\) \(2.551751385\)
\(L(\frac12)\) \(\approx\) \(2.551751385\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.690T + 8T^{2} \)
5 \( 1 - 4.55T + 125T^{2} \)
7 \( 1 - 26.5T + 343T^{2} \)
13 \( 1 - 14.5T + 2.19e3T^{2} \)
17 \( 1 - 68.2T + 4.91e3T^{2} \)
19 \( 1 - 139.T + 6.85e3T^{2} \)
23 \( 1 - 17.3T + 1.21e4T^{2} \)
29 \( 1 + 160.T + 2.43e4T^{2} \)
31 \( 1 - 241.T + 2.97e4T^{2} \)
37 \( 1 - 307.T + 5.06e4T^{2} \)
41 \( 1 - 316.T + 6.89e4T^{2} \)
43 \( 1 - 290.T + 7.95e4T^{2} \)
47 \( 1 + 131.T + 1.03e5T^{2} \)
53 \( 1 + 163.T + 1.48e5T^{2} \)
59 \( 1 + 261.T + 2.05e5T^{2} \)
67 \( 1 + 996.T + 3.00e5T^{2} \)
71 \( 1 + 53.1T + 3.57e5T^{2} \)
73 \( 1 + 143.T + 3.89e5T^{2} \)
79 \( 1 + 529.T + 4.93e5T^{2} \)
83 \( 1 + 1.45e3T + 5.71e5T^{2} \)
89 \( 1 + 987.T + 7.04e5T^{2} \)
97 \( 1 - 1.26e3T + 9.12e5T^{2} \)
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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

−8.867747591170626114938762238248, −7.84620747063671053718883929221, −7.52046572305246120470979030937, −5.92222820027727232901407360420, −5.67478425317837964895384730603, −4.75157438789250417745740782790, −4.17092213177188251427411013014, −3.01888184084603836814253847234, −1.49521096992562817880073322856, −0.836913857490103003999332175457, 0.836913857490103003999332175457, 1.49521096992562817880073322856, 3.01888184084603836814253847234, 4.17092213177188251427411013014, 4.75157438789250417745740782790, 5.67478425317837964895384730603, 5.92222820027727232901407360420, 7.52046572305246120470979030937, 7.84620747063671053718883929221, 8.867747591170626114938762238248

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