Properties

Label 2-2013-1.1-c1-0-64
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  − 1.45·2-s + 3-s + 0.120·4-s − 0.271·5-s − 1.45·6-s − 0.415·7-s + 2.73·8-s + 9-s + 0.395·10-s + 11-s + 0.120·12-s − 2.31·13-s + 0.604·14-s − 0.271·15-s − 4.22·16-s − 4.82·17-s − 1.45·18-s + 3.21·19-s − 0.0328·20-s − 0.415·21-s − 1.45·22-s + 4.64·23-s + 2.73·24-s − 4.92·25-s + 3.37·26-s + 27-s − 0.0502·28-s + ⋯
L(s)  = 1  − 1.02·2-s + 0.577·3-s + 0.0604·4-s − 0.121·5-s − 0.594·6-s − 0.156·7-s + 0.967·8-s + 0.333·9-s + 0.124·10-s + 0.301·11-s + 0.0348·12-s − 0.642·13-s + 0.161·14-s − 0.0700·15-s − 1.05·16-s − 1.17·17-s − 0.343·18-s + 0.736·19-s − 0.00733·20-s − 0.0906·21-s − 0.310·22-s + 0.969·23-s + 0.558·24-s − 0.985·25-s + 0.661·26-s + 0.192·27-s − 0.00948·28-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 \( 1 + 1.45T + 2T^{2} \)
5 \( 1 + 0.271T + 5T^{2} \)
7 \( 1 + 0.415T + 7T^{2} \)
13 \( 1 + 2.31T + 13T^{2} \)
17 \( 1 + 4.82T + 17T^{2} \)
19 \( 1 - 3.21T + 19T^{2} \)
23 \( 1 - 4.64T + 23T^{2} \)
29 \( 1 + 7.77T + 29T^{2} \)
31 \( 1 + 6.54T + 31T^{2} \)
37 \( 1 - 9.79T + 37T^{2} \)
41 \( 1 + 5.80T + 41T^{2} \)
43 \( 1 - 8.50T + 43T^{2} \)
47 \( 1 - 4.52T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 - 8.42T + 59T^{2} \)
67 \( 1 + 2.02T + 67T^{2} \)
71 \( 1 - 1.09T + 71T^{2} \)
73 \( 1 + 2.43T + 73T^{2} \)
79 \( 1 - 7.63T + 79T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
89 \( 1 + 8.01T + 89T^{2} \)
97 \( 1 + 3.94T + 97T^{2} \)
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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

−8.967306068454221873508433535881, −8.031756242932080656848193551912, −7.45017882175339253880340841685, −6.81951006027511217246201845870, −5.56083460160577614906979148713, −4.54431660245095920523530490020, −3.76384878606675420802902274045, −2.52035121675953478872172692671, −1.48556400872453263146757795408, 0, 1.48556400872453263146757795408, 2.52035121675953478872172692671, 3.76384878606675420802902274045, 4.54431660245095920523530490020, 5.56083460160577614906979148713, 6.81951006027511217246201845870, 7.45017882175339253880340841685, 8.031756242932080656848193551912, 8.967306068454221873508433535881

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