Properties

Label 2-2013-1.1-c1-0-44
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.05·2-s − 3-s − 0.883·4-s + 1.53·5-s + 1.05·6-s − 3.09·7-s + 3.04·8-s + 9-s − 1.61·10-s − 11-s + 0.883·12-s + 3.38·13-s + 3.26·14-s − 1.53·15-s − 1.45·16-s − 3.57·17-s − 1.05·18-s + 1.31·19-s − 1.35·20-s + 3.09·21-s + 1.05·22-s − 5.12·23-s − 3.04·24-s − 2.65·25-s − 3.57·26-s − 27-s + 2.73·28-s + ⋯
L(s)  = 1  − 0.747·2-s − 0.577·3-s − 0.441·4-s + 0.684·5-s + 0.431·6-s − 1.16·7-s + 1.07·8-s + 0.333·9-s − 0.511·10-s − 0.301·11-s + 0.254·12-s + 0.938·13-s + 0.873·14-s − 0.395·15-s − 0.363·16-s − 0.867·17-s − 0.249·18-s + 0.301·19-s − 0.302·20-s + 0.674·21-s + 0.225·22-s − 1.06·23-s − 0.621·24-s − 0.531·25-s − 0.701·26-s − 0.192·27-s + 0.516·28-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 \( 1 + 1.05T + 2T^{2} \)
5 \( 1 - 1.53T + 5T^{2} \)
7 \( 1 + 3.09T + 7T^{2} \)
13 \( 1 - 3.38T + 13T^{2} \)
17 \( 1 + 3.57T + 17T^{2} \)
19 \( 1 - 1.31T + 19T^{2} \)
23 \( 1 + 5.12T + 23T^{2} \)
29 \( 1 - 4.88T + 29T^{2} \)
31 \( 1 - 6.01T + 31T^{2} \)
37 \( 1 - 5.49T + 37T^{2} \)
41 \( 1 - 8.30T + 41T^{2} \)
43 \( 1 + 6.40T + 43T^{2} \)
47 \( 1 - 4.95T + 47T^{2} \)
53 \( 1 - 1.09T + 53T^{2} \)
59 \( 1 + 4.40T + 59T^{2} \)
67 \( 1 - 3.91T + 67T^{2} \)
71 \( 1 + 6.45T + 71T^{2} \)
73 \( 1 + 9.63T + 73T^{2} \)
79 \( 1 + 2.79T + 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.869939492482547282690486857261, −8.171124547088595776216638568880, −7.20587331172839362782326899784, −6.22715933256345311458082516261, −5.89437431419828095568862102184, −4.67774699315458175970810449131, −3.89462934865076389730004165026, −2.58734668803053703886646186341, −1.25231657983292375202851797777, 0, 1.25231657983292375202851797777, 2.58734668803053703886646186341, 3.89462934865076389730004165026, 4.67774699315458175970810449131, 5.89437431419828095568862102184, 6.22715933256345311458082516261, 7.20587331172839362782326899784, 8.171124547088595776216638568880, 8.869939492482547282690486857261

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