Properties

Label 2-2013-1.1-c3-0-59
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  − 4.08·2-s + 3·3-s + 8.72·4-s − 14.1·5-s − 12.2·6-s − 18.6·7-s − 2.97·8-s + 9·9-s + 57.9·10-s + 11·11-s + 26.1·12-s + 63.5·13-s + 76.3·14-s − 42.5·15-s − 57.6·16-s + 65.4·17-s − 36.8·18-s − 27.2·19-s − 123.·20-s − 56.0·21-s − 44.9·22-s + 43.8·23-s − 8.91·24-s + 75.7·25-s − 259.·26-s + 27·27-s − 162.·28-s + ⋯
L(s)  = 1  − 1.44·2-s + 0.577·3-s + 1.09·4-s − 1.26·5-s − 0.834·6-s − 1.00·7-s − 0.131·8-s + 0.333·9-s + 1.83·10-s + 0.301·11-s + 0.629·12-s + 1.35·13-s + 1.45·14-s − 0.731·15-s − 0.900·16-s + 0.934·17-s − 0.481·18-s − 0.329·19-s − 1.38·20-s − 0.581·21-s − 0.435·22-s + 0.397·23-s − 0.0758·24-s + 0.605·25-s − 1.95·26-s + 0.192·27-s − 1.09·28-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

\(L(2)\) \(\approx\) \(0.7819217847\)
\(L(\frac12)\) \(\approx\) \(0.7819217847\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.08T + 8T^{2} \)
5 \( 1 + 14.1T + 125T^{2} \)
7 \( 1 + 18.6T + 343T^{2} \)
13 \( 1 - 63.5T + 2.19e3T^{2} \)
17 \( 1 - 65.4T + 4.91e3T^{2} \)
19 \( 1 + 27.2T + 6.85e3T^{2} \)
23 \( 1 - 43.8T + 1.21e4T^{2} \)
29 \( 1 + 169.T + 2.43e4T^{2} \)
31 \( 1 - 341.T + 2.97e4T^{2} \)
37 \( 1 - 6.98T + 5.06e4T^{2} \)
41 \( 1 - 255.T + 6.89e4T^{2} \)
43 \( 1 + 301.T + 7.95e4T^{2} \)
47 \( 1 - 83.8T + 1.03e5T^{2} \)
53 \( 1 - 260.T + 1.48e5T^{2} \)
59 \( 1 + 66.5T + 2.05e5T^{2} \)
67 \( 1 + 443.T + 3.00e5T^{2} \)
71 \( 1 + 482.T + 3.57e5T^{2} \)
73 \( 1 + 177.T + 3.89e5T^{2} \)
79 \( 1 - 211.T + 4.93e5T^{2} \)
83 \( 1 + 948.T + 5.71e5T^{2} \)
89 \( 1 - 680.T + 7.04e5T^{2} \)
97 \( 1 + 1.47e3T + 9.12e5T^{2} \)
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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

−8.676639281255520651023333568544, −8.215285141423364791815632443745, −7.53660170836948272551638073883, −6.82800950451674832558820120384, −5.99845612738679705446100149250, −4.40596514424173068021005599016, −3.65184900808710959552527623359, −2.87635100828853936706817250250, −1.41215980518246048436209557348, −0.53699641831291727239177828288, 0.53699641831291727239177828288, 1.41215980518246048436209557348, 2.87635100828853936706817250250, 3.65184900808710959552527623359, 4.40596514424173068021005599016, 5.99845612738679705446100149250, 6.82800950451674832558820120384, 7.53660170836948272551638073883, 8.215285141423364791815632443745, 8.676639281255520651023333568544

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