Properties

Label 2-2013-1.1-c1-0-57
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  − 1.72·2-s − 3-s + 0.973·4-s − 0.0158·5-s + 1.72·6-s + 3.08·7-s + 1.77·8-s + 9-s + 0.0273·10-s − 11-s − 0.973·12-s + 0.101·13-s − 5.32·14-s + 0.0158·15-s − 4.99·16-s − 0.754·17-s − 1.72·18-s − 6.01·19-s − 0.0154·20-s − 3.08·21-s + 1.72·22-s + 7.29·23-s − 1.77·24-s − 4.99·25-s − 0.175·26-s − 27-s + 3.00·28-s + ⋯
L(s)  = 1  − 1.21·2-s − 0.577·3-s + 0.486·4-s − 0.00710·5-s + 0.703·6-s + 1.16·7-s + 0.625·8-s + 0.333·9-s + 0.00865·10-s − 0.301·11-s − 0.280·12-s + 0.0281·13-s − 1.42·14-s + 0.00409·15-s − 1.24·16-s − 0.182·17-s − 0.406·18-s − 1.37·19-s − 0.00345·20-s − 0.673·21-s + 0.367·22-s + 1.52·23-s − 0.361·24-s − 0.999·25-s − 0.0343·26-s − 0.192·27-s + 0.568·28-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 \( 1 + 1.72T + 2T^{2} \)
5 \( 1 + 0.0158T + 5T^{2} \)
7 \( 1 - 3.08T + 7T^{2} \)
13 \( 1 - 0.101T + 13T^{2} \)
17 \( 1 + 0.754T + 17T^{2} \)
19 \( 1 + 6.01T + 19T^{2} \)
23 \( 1 - 7.29T + 23T^{2} \)
29 \( 1 - 2.30T + 29T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + 1.17T + 37T^{2} \)
41 \( 1 + 6.03T + 41T^{2} \)
43 \( 1 + 3.46T + 43T^{2} \)
47 \( 1 - 4.51T + 47T^{2} \)
53 \( 1 - 5.67T + 53T^{2} \)
59 \( 1 - 1.22T + 59T^{2} \)
67 \( 1 - 9.40T + 67T^{2} \)
71 \( 1 + 9.66T + 71T^{2} \)
73 \( 1 - 5.87T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 - 5.44T + 89T^{2} \)
97 \( 1 + 6.37T + 97T^{2} \)
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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

−8.682713254383431353740653386500, −8.179889131619529097146950590057, −7.34151095951755057227216493122, −6.71767751201858531820739852743, −5.48209149745464779920654747319, −4.82462860884360394012457378939, −3.93936225712806281070654525522, −2.21825798597905997329159706244, −1.36511887351668887794095117023, 0, 1.36511887351668887794095117023, 2.21825798597905997329159706244, 3.93936225712806281070654525522, 4.82462860884360394012457378939, 5.48209149745464779920654747319, 6.71767751201858531820739852743, 7.34151095951755057227216493122, 8.179889131619529097146950590057, 8.682713254383431353740653386500

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