Properties

Label 2-2013-1.1-c3-0-121
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  − 2.21·2-s − 3·3-s − 3.08·4-s − 20.1·5-s + 6.64·6-s + 13.2·7-s + 24.5·8-s + 9·9-s + 44.6·10-s − 11·11-s + 9.26·12-s + 59.5·13-s − 29.2·14-s + 60.4·15-s − 29.7·16-s + 67.8·17-s − 19.9·18-s + 19.2·19-s + 62.2·20-s − 39.6·21-s + 24.3·22-s + 112.·23-s − 73.7·24-s + 281.·25-s − 131.·26-s − 27·27-s − 40.8·28-s + ⋯
L(s)  = 1  − 0.783·2-s − 0.577·3-s − 0.385·4-s − 1.80·5-s + 0.452·6-s + 0.713·7-s + 1.08·8-s + 0.333·9-s + 1.41·10-s − 0.301·11-s + 0.222·12-s + 1.27·13-s − 0.559·14-s + 1.04·15-s − 0.465·16-s + 0.968·17-s − 0.261·18-s + 0.232·19-s + 0.696·20-s − 0.411·21-s + 0.236·22-s + 1.01·23-s − 0.627·24-s + 2.25·25-s − 0.995·26-s − 0.192·27-s − 0.275·28-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

\(L(2)\) \(\approx\) \(1.055678917\)
\(L(\frac12)\) \(\approx\) \(1.055678917\)
\(L(\frac{5}{2})\) 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 \( 1 + 2.21T + 8T^{2} \)
5 \( 1 + 20.1T + 125T^{2} \)
7 \( 1 - 13.2T + 343T^{2} \)
13 \( 1 - 59.5T + 2.19e3T^{2} \)
17 \( 1 - 67.8T + 4.91e3T^{2} \)
19 \( 1 - 19.2T + 6.85e3T^{2} \)
23 \( 1 - 112.T + 1.21e4T^{2} \)
29 \( 1 - 214.T + 2.43e4T^{2} \)
31 \( 1 - 235.T + 2.97e4T^{2} \)
37 \( 1 - 336.T + 5.06e4T^{2} \)
41 \( 1 - 130.T + 6.89e4T^{2} \)
43 \( 1 - 466.T + 7.95e4T^{2} \)
47 \( 1 - 71.5T + 1.03e5T^{2} \)
53 \( 1 - 390.T + 1.48e5T^{2} \)
59 \( 1 - 19.9T + 2.05e5T^{2} \)
67 \( 1 + 206.T + 3.00e5T^{2} \)
71 \( 1 - 206.T + 3.57e5T^{2} \)
73 \( 1 - 757.T + 3.89e5T^{2} \)
79 \( 1 - 118.T + 4.93e5T^{2} \)
83 \( 1 + 613.T + 5.71e5T^{2} \)
89 \( 1 - 434.T + 7.04e5T^{2} \)
97 \( 1 + 981.T + 9.12e5T^{2} \)
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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

−8.534049199636678277325850111132, −8.053735371473422500007308288074, −7.61506214356497439426958081427, −6.69003109781943394427250288116, −5.48345487842022665934324097171, −4.54342597944597906181598263151, −4.12311627119401459193111221668, −3.00431067630835784816297982063, −0.968458871822237505014973207817, −0.823852243193761481877916519310, 0.823852243193761481877916519310, 0.968458871822237505014973207817, 3.00431067630835784816297982063, 4.12311627119401459193111221668, 4.54342597944597906181598263151, 5.48345487842022665934324097171, 6.69003109781943394427250288116, 7.61506214356497439426958081427, 8.053735371473422500007308288074, 8.534049199636678277325850111132

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