Properties

Label 2-2013-1.1-c3-0-299
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.16·2-s + 3·3-s + 9.37·4-s + 18.6·5-s + 12.5·6-s − 32.7·7-s + 5.72·8-s + 9·9-s + 77.5·10-s + 11·11-s + 28.1·12-s − 88.0·13-s − 136.·14-s + 55.8·15-s − 51.1·16-s − 71.0·17-s + 37.5·18-s − 15.2·19-s + 174.·20-s − 98.2·21-s + 45.8·22-s − 103.·23-s + 17.1·24-s + 221.·25-s − 366.·26-s + 27·27-s − 307.·28-s + ⋯
L(s)  = 1  + 1.47·2-s + 0.577·3-s + 1.17·4-s + 1.66·5-s + 0.850·6-s − 1.76·7-s + 0.253·8-s + 0.333·9-s + 2.45·10-s + 0.301·11-s + 0.676·12-s − 1.87·13-s − 2.60·14-s + 0.961·15-s − 0.798·16-s − 1.01·17-s + 0.491·18-s − 0.183·19-s + 1.95·20-s − 1.02·21-s + 0.444·22-s − 0.938·23-s + 0.146·24-s + 1.77·25-s − 2.76·26-s + 0.192·27-s − 2.07·28-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: \(1\)
Selberg data: \((2,\ 2013,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.16T + 8T^{2} \)
5 \( 1 - 18.6T + 125T^{2} \)
7 \( 1 + 32.7T + 343T^{2} \)
13 \( 1 + 88.0T + 2.19e3T^{2} \)
17 \( 1 + 71.0T + 4.91e3T^{2} \)
19 \( 1 + 15.2T + 6.85e3T^{2} \)
23 \( 1 + 103.T + 1.21e4T^{2} \)
29 \( 1 - 177.T + 2.43e4T^{2} \)
31 \( 1 - 305.T + 2.97e4T^{2} \)
37 \( 1 + 369.T + 5.06e4T^{2} \)
41 \( 1 + 354.T + 6.89e4T^{2} \)
43 \( 1 - 198.T + 7.95e4T^{2} \)
47 \( 1 + 399.T + 1.03e5T^{2} \)
53 \( 1 + 190.T + 1.48e5T^{2} \)
59 \( 1 + 88.2T + 2.05e5T^{2} \)
67 \( 1 + 596.T + 3.00e5T^{2} \)
71 \( 1 - 804.T + 3.57e5T^{2} \)
73 \( 1 + 498.T + 3.89e5T^{2} \)
79 \( 1 + 1.03e3T + 4.93e5T^{2} \)
83 \( 1 - 358.T + 5.71e5T^{2} \)
89 \( 1 + 1.36e3T + 7.04e5T^{2} \)
97 \( 1 - 1.10e3T + 9.12e5T^{2} \)
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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

−8.629883220688277100338862848892, −7.03827735167383523179733574087, −6.54771803942598513137819384457, −6.09291844969471314806772813928, −5.08101332727873565972420517300, −4.41443775671141898571591635792, −3.20417154160281177346119094399, −2.65913428017054737101600457178, −1.98378645146303187545008981224, 0, 1.98378645146303187545008981224, 2.65913428017054737101600457178, 3.20417154160281177346119094399, 4.41443775671141898571591635792, 5.08101332727873565972420517300, 6.09291844969471314806772813928, 6.54771803942598513137819384457, 7.03827735167383523179733574087, 8.629883220688277100338862848892

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