Properties

Label 2-2013-1.1-c1-0-50
Degree $2$
Conductor $2013$
Sign $-1$
Analytic cond. $16.0738$
Root an. cond. $4.00922$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.70·2-s − 3-s + 5.29·4-s + 2.21·5-s + 2.70·6-s − 2.11·7-s − 8.88·8-s + 9-s − 5.99·10-s + 11-s − 5.29·12-s − 2.52·13-s + 5.72·14-s − 2.21·15-s + 13.4·16-s + 1.86·17-s − 2.70·18-s − 1.62·19-s + 11.7·20-s + 2.11·21-s − 2.70·22-s + 0.232·23-s + 8.88·24-s − 0.0735·25-s + 6.81·26-s − 27-s − 11.2·28-s + ⋯
L(s)  = 1  − 1.90·2-s − 0.577·3-s + 2.64·4-s + 0.992·5-s + 1.10·6-s − 0.800·7-s − 3.14·8-s + 0.333·9-s − 1.89·10-s + 0.301·11-s − 1.52·12-s − 0.699·13-s + 1.52·14-s − 0.573·15-s + 3.35·16-s + 0.451·17-s − 0.636·18-s − 0.372·19-s + 2.62·20-s + 0.462·21-s − 0.575·22-s + 0.0484·23-s + 1.81·24-s − 0.0147·25-s + 1.33·26-s − 0.192·27-s − 2.11·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-1$
Analytic conductor: \(16.0738\)
Root analytic conductor: \(4.00922\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2013,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
11 \( 1 - T \)
61 \( 1 - T \)
good2 \( 1 + 2.70T + 2T^{2} \)
5 \( 1 - 2.21T + 5T^{2} \)
7 \( 1 + 2.11T + 7T^{2} \)
13 \( 1 + 2.52T + 13T^{2} \)
17 \( 1 - 1.86T + 17T^{2} \)
19 \( 1 + 1.62T + 19T^{2} \)
23 \( 1 - 0.232T + 23T^{2} \)
29 \( 1 - 8.99T + 29T^{2} \)
31 \( 1 + 0.0529T + 31T^{2} \)
37 \( 1 + 1.12T + 37T^{2} \)
41 \( 1 + 8.95T + 41T^{2} \)
43 \( 1 + 8.83T + 43T^{2} \)
47 \( 1 + 8.15T + 47T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 + 1.89T + 59T^{2} \)
67 \( 1 + 1.07T + 67T^{2} \)
71 \( 1 - 5.42T + 71T^{2} \)
73 \( 1 + 8.03T + 73T^{2} \)
79 \( 1 + 6.67T + 79T^{2} \)
83 \( 1 - 3.06T + 83T^{2} \)
89 \( 1 - 5.40T + 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−8.874922598229526053124892826825, −8.208622536346452072552261229195, −7.15707717609761415513693795400, −6.57526995671263749289081490924, −6.07550643459120680797908633825, −5.04026660607631643662333639557, −3.28410550009955990852021099286, −2.27722901705640007760320447445, −1.31145948635138763594681064837, 0, 1.31145948635138763594681064837, 2.27722901705640007760320447445, 3.28410550009955990852021099286, 5.04026660607631643662333639557, 6.07550643459120680797908633825, 6.57526995671263749289081490924, 7.15707717609761415513693795400, 8.208622536346452072552261229195, 8.874922598229526053124892826825

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