Properties

Label 2-2013-1.1-c3-0-142
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  + 4.78·2-s − 3·3-s + 14.9·4-s − 3.84·5-s − 14.3·6-s + 23.4·7-s + 33.1·8-s + 9·9-s − 18.3·10-s + 11·11-s − 44.7·12-s − 80.3·13-s + 112.·14-s + 11.5·15-s + 39.4·16-s + 84.9·17-s + 43.0·18-s + 10.7·19-s − 57.3·20-s − 70.4·21-s + 52.6·22-s + 148.·23-s − 99.5·24-s − 110.·25-s − 384.·26-s − 27·27-s + 350.·28-s + ⋯
L(s)  = 1  + 1.69·2-s − 0.577·3-s + 1.86·4-s − 0.343·5-s − 0.977·6-s + 1.26·7-s + 1.46·8-s + 0.333·9-s − 0.581·10-s + 0.301·11-s − 1.07·12-s − 1.71·13-s + 2.14·14-s + 0.198·15-s + 0.617·16-s + 1.21·17-s + 0.564·18-s + 0.129·19-s − 0.641·20-s − 0.732·21-s + 0.510·22-s + 1.34·23-s − 0.846·24-s − 0.881·25-s − 2.90·26-s − 0.192·27-s + 2.36·28-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

\(L(2)\) \(\approx\) \(6.000907107\)
\(L(\frac12)\) \(\approx\) \(6.000907107\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.78T + 8T^{2} \)
5 \( 1 + 3.84T + 125T^{2} \)
7 \( 1 - 23.4T + 343T^{2} \)
13 \( 1 + 80.3T + 2.19e3T^{2} \)
17 \( 1 - 84.9T + 4.91e3T^{2} \)
19 \( 1 - 10.7T + 6.85e3T^{2} \)
23 \( 1 - 148.T + 1.21e4T^{2} \)
29 \( 1 - 178.T + 2.43e4T^{2} \)
31 \( 1 - 185.T + 2.97e4T^{2} \)
37 \( 1 - 242.T + 5.06e4T^{2} \)
41 \( 1 + 518.T + 6.89e4T^{2} \)
43 \( 1 + 129.T + 7.95e4T^{2} \)
47 \( 1 - 495.T + 1.03e5T^{2} \)
53 \( 1 - 557.T + 1.48e5T^{2} \)
59 \( 1 - 498.T + 2.05e5T^{2} \)
67 \( 1 + 456.T + 3.00e5T^{2} \)
71 \( 1 + 705.T + 3.57e5T^{2} \)
73 \( 1 - 581.T + 3.89e5T^{2} \)
79 \( 1 + 566.T + 4.93e5T^{2} \)
83 \( 1 - 748.T + 5.71e5T^{2} \)
89 \( 1 - 497.T + 7.04e5T^{2} \)
97 \( 1 - 591.T + 9.12e5T^{2} \)
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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

−8.586805624408686458315715732406, −7.56460639218085532177384561985, −7.14203302106387722945195627166, −6.14566294486444620710824246856, −5.19404441823632328133802459969, −4.90659212204329507267732601352, −4.18077240987340612542194801600, −3.09881874540743441395135778677, −2.15827698141063325454878287824, −0.923005848573767945589638088114, 0.923005848573767945589638088114, 2.15827698141063325454878287824, 3.09881874540743441395135778677, 4.18077240987340612542194801600, 4.90659212204329507267732601352, 5.19404441823632328133802459969, 6.14566294486444620710824246856, 7.14203302106387722945195627166, 7.56460639218085532177384561985, 8.586805624408686458315715732406

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