Properties

Label 2-2013-1.1-c3-0-60
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  − 3.67·2-s + 3·3-s + 5.47·4-s + 14.7·5-s − 11.0·6-s − 28.2·7-s + 9.26·8-s + 9·9-s − 54.1·10-s − 11·11-s + 16.4·12-s − 27.9·13-s + 103.·14-s + 44.2·15-s − 77.8·16-s − 25.0·17-s − 33.0·18-s + 89.2·19-s + 80.7·20-s − 84.6·21-s + 40.3·22-s − 170.·23-s + 27.7·24-s + 92.2·25-s + 102.·26-s + 27·27-s − 154.·28-s + ⋯
L(s)  = 1  − 1.29·2-s + 0.577·3-s + 0.684·4-s + 1.31·5-s − 0.749·6-s − 1.52·7-s + 0.409·8-s + 0.333·9-s − 1.71·10-s − 0.301·11-s + 0.395·12-s − 0.595·13-s + 1.97·14-s + 0.761·15-s − 1.21·16-s − 0.357·17-s − 0.432·18-s + 1.07·19-s + 0.902·20-s − 0.879·21-s + 0.391·22-s − 1.54·23-s + 0.236·24-s + 0.737·25-s + 0.772·26-s + 0.192·27-s − 1.04·28-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

\(L(2)\) \(\approx\) \(1.042863144\)
\(L(\frac12)\) \(\approx\) \(1.042863144\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.67T + 8T^{2} \)
5 \( 1 - 14.7T + 125T^{2} \)
7 \( 1 + 28.2T + 343T^{2} \)
13 \( 1 + 27.9T + 2.19e3T^{2} \)
17 \( 1 + 25.0T + 4.91e3T^{2} \)
19 \( 1 - 89.2T + 6.85e3T^{2} \)
23 \( 1 + 170.T + 1.21e4T^{2} \)
29 \( 1 - 61.6T + 2.43e4T^{2} \)
31 \( 1 - 171.T + 2.97e4T^{2} \)
37 \( 1 + 110.T + 5.06e4T^{2} \)
41 \( 1 + 438.T + 6.89e4T^{2} \)
43 \( 1 + 325.T + 7.95e4T^{2} \)
47 \( 1 - 390.T + 1.03e5T^{2} \)
53 \( 1 + 136.T + 1.48e5T^{2} \)
59 \( 1 - 552.T + 2.05e5T^{2} \)
67 \( 1 - 802.T + 3.00e5T^{2} \)
71 \( 1 + 823.T + 3.57e5T^{2} \)
73 \( 1 - 695.T + 3.89e5T^{2} \)
79 \( 1 + 1.20e3T + 4.93e5T^{2} \)
83 \( 1 + 73.7T + 5.71e5T^{2} \)
89 \( 1 - 996.T + 7.04e5T^{2} \)
97 \( 1 - 313.T + 9.12e5T^{2} \)
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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

−8.917594712369580244450608787859, −8.303219134586501653769143291448, −7.30237314307590657509329718813, −6.67887556045106656119004815204, −5.88935932203690093785586561711, −4.83833655044360538247470981913, −3.51147283175753714863179951505, −2.52903411624288412860256564444, −1.80889430389939835970775123289, −0.54193481384584403094593924494, 0.54193481384584403094593924494, 1.80889430389939835970775123289, 2.52903411624288412860256564444, 3.51147283175753714863179951505, 4.83833655044360538247470981913, 5.88935932203690093785586561711, 6.67887556045106656119004815204, 7.30237314307590657509329718813, 8.303219134586501653769143291448, 8.917594712369580244450608787859

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