Properties

Label 2-2013-1.1-c3-0-83
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  − 4.92·2-s − 3·3-s + 16.2·4-s + 10.2·5-s + 14.7·6-s + 17.4·7-s − 40.7·8-s + 9·9-s − 50.5·10-s + 11·11-s − 48.8·12-s + 72.1·13-s − 86.0·14-s − 30.7·15-s + 70.6·16-s − 57.0·17-s − 44.3·18-s + 4.76·19-s + 166.·20-s − 52.3·21-s − 54.1·22-s − 122.·23-s + 122.·24-s − 19.9·25-s − 355.·26-s − 27·27-s + 284.·28-s + ⋯
L(s)  = 1  − 1.74·2-s − 0.577·3-s + 2.03·4-s + 0.916·5-s + 1.00·6-s + 0.942·7-s − 1.80·8-s + 0.333·9-s − 1.59·10-s + 0.301·11-s − 1.17·12-s + 1.53·13-s − 1.64·14-s − 0.529·15-s + 1.10·16-s − 0.813·17-s − 0.580·18-s + 0.0575·19-s + 1.86·20-s − 0.544·21-s − 0.525·22-s − 1.11·23-s + 1.04·24-s − 0.159·25-s − 2.68·26-s − 0.192·27-s + 1.91·28-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

\(L(2)\) \(\approx\) \(1.060342251\)
\(L(\frac12)\) \(\approx\) \(1.060342251\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.92T + 8T^{2} \)
5 \( 1 - 10.2T + 125T^{2} \)
7 \( 1 - 17.4T + 343T^{2} \)
13 \( 1 - 72.1T + 2.19e3T^{2} \)
17 \( 1 + 57.0T + 4.91e3T^{2} \)
19 \( 1 - 4.76T + 6.85e3T^{2} \)
23 \( 1 + 122.T + 1.21e4T^{2} \)
29 \( 1 + 148.T + 2.43e4T^{2} \)
31 \( 1 - 154.T + 2.97e4T^{2} \)
37 \( 1 + 86.1T + 5.06e4T^{2} \)
41 \( 1 + 129.T + 6.89e4T^{2} \)
43 \( 1 + 394.T + 7.95e4T^{2} \)
47 \( 1 - 18.9T + 1.03e5T^{2} \)
53 \( 1 - 367.T + 1.48e5T^{2} \)
59 \( 1 - 180.T + 2.05e5T^{2} \)
67 \( 1 - 833.T + 3.00e5T^{2} \)
71 \( 1 + 411.T + 3.57e5T^{2} \)
73 \( 1 - 383.T + 3.89e5T^{2} \)
79 \( 1 - 671.T + 4.93e5T^{2} \)
83 \( 1 + 729.T + 5.71e5T^{2} \)
89 \( 1 - 1.38e3T + 7.04e5T^{2} \)
97 \( 1 - 1.75e3T + 9.12e5T^{2} \)
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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

−8.691214620358825221924343410229, −8.331184963360185217174732901848, −7.41092773885450091658475540539, −6.44157651383689009812940724527, −6.06444793462037309931561908724, −4.99009928187275964309000333456, −3.74400714086241489918873487492, −2.06999683417935053953464610950, −1.66078672485626543213202976681, −0.66200264823025723591960171986, 0.66200264823025723591960171986, 1.66078672485626543213202976681, 2.06999683417935053953464610950, 3.74400714086241489918873487492, 4.99009928187275964309000333456, 6.06444793462037309931561908724, 6.44157651383689009812940724527, 7.41092773885450091658475540539, 8.331184963360185217174732901848, 8.691214620358825221924343410229

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