Properties

Label 2-2013-1.1-c3-0-35
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  + 2.46·2-s + 3·3-s − 1.90·4-s − 6.00·5-s + 7.40·6-s − 34.9·7-s − 24.4·8-s + 9·9-s − 14.8·10-s + 11·11-s − 5.71·12-s + 13.1·13-s − 86.3·14-s − 18.0·15-s − 45.1·16-s + 9.32·17-s + 22.2·18-s − 134.·19-s + 11.4·20-s − 104.·21-s + 27.1·22-s − 30.9·23-s − 73.3·24-s − 88.9·25-s + 32.5·26-s + 27·27-s + 66.6·28-s + ⋯
L(s)  = 1  + 0.872·2-s + 0.577·3-s − 0.238·4-s − 0.537·5-s + 0.503·6-s − 1.88·7-s − 1.08·8-s + 0.333·9-s − 0.468·10-s + 0.301·11-s − 0.137·12-s + 0.280·13-s − 1.64·14-s − 0.310·15-s − 0.705·16-s + 0.132·17-s + 0.290·18-s − 1.62·19-s + 0.127·20-s − 1.09·21-s + 0.263·22-s − 0.280·23-s − 0.623·24-s − 0.711·25-s + 0.245·26-s + 0.192·27-s + 0.449·28-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

\(L(2)\) \(\approx\) \(1.196222206\)
\(L(\frac12)\) \(\approx\) \(1.196222206\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.46T + 8T^{2} \)
5 \( 1 + 6.00T + 125T^{2} \)
7 \( 1 + 34.9T + 343T^{2} \)
13 \( 1 - 13.1T + 2.19e3T^{2} \)
17 \( 1 - 9.32T + 4.91e3T^{2} \)
19 \( 1 + 134.T + 6.85e3T^{2} \)
23 \( 1 + 30.9T + 1.21e4T^{2} \)
29 \( 1 + 97.1T + 2.43e4T^{2} \)
31 \( 1 - 8.42T + 2.97e4T^{2} \)
37 \( 1 + 217.T + 5.06e4T^{2} \)
41 \( 1 - 125.T + 6.89e4T^{2} \)
43 \( 1 - 379.T + 7.95e4T^{2} \)
47 \( 1 + 335.T + 1.03e5T^{2} \)
53 \( 1 + 134.T + 1.48e5T^{2} \)
59 \( 1 - 501.T + 2.05e5T^{2} \)
67 \( 1 - 241.T + 3.00e5T^{2} \)
71 \( 1 + 630.T + 3.57e5T^{2} \)
73 \( 1 + 19.4T + 3.89e5T^{2} \)
79 \( 1 - 422.T + 4.93e5T^{2} \)
83 \( 1 - 973.T + 5.71e5T^{2} \)
89 \( 1 - 292.T + 7.04e5T^{2} \)
97 \( 1 - 755.T + 9.12e5T^{2} \)
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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

−8.897160383941089740229595101249, −8.106079945784636661486314896371, −7.00595309662436087770642687447, −6.32609062404393147201219949934, −5.70323826947663029926263350013, −4.38089007749598282512840269350, −3.79078886477407775151614869520, −3.27768919574084366292550856273, −2.26481692782958062737958367015, −0.40205850927152692261705610458, 0.40205850927152692261705610458, 2.26481692782958062737958367015, 3.27768919574084366292550856273, 3.79078886477407775151614869520, 4.38089007749598282512840269350, 5.70323826947663029926263350013, 6.32609062404393147201219949934, 7.00595309662436087770642687447, 8.106079945784636661486314896371, 8.897160383941089740229595101249

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