Properties

Label 2-2013-1.1-c3-0-79
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 $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.789·2-s − 3·3-s − 7.37·4-s + 13.9·5-s + 2.36·6-s − 21.1·7-s + 12.1·8-s + 9·9-s − 10.9·10-s − 11·11-s + 22.1·12-s + 41.9·13-s + 16.6·14-s − 41.7·15-s + 49.4·16-s + 53.5·17-s − 7.10·18-s + 133.·19-s − 102.·20-s + 63.4·21-s + 8.68·22-s − 11.7·23-s − 36.4·24-s + 69.0·25-s − 33.1·26-s − 27·27-s + 156.·28-s + ⋯
L(s)  = 1  − 0.279·2-s − 0.577·3-s − 0.922·4-s + 1.24·5-s + 0.161·6-s − 1.14·7-s + 0.536·8-s + 0.333·9-s − 0.347·10-s − 0.301·11-s + 0.532·12-s + 0.895·13-s + 0.318·14-s − 0.719·15-s + 0.772·16-s + 0.763·17-s − 0.0930·18-s + 1.61·19-s − 1.14·20-s + 0.659·21-s + 0.0841·22-s − 0.106·23-s − 0.309·24-s + 0.552·25-s − 0.249·26-s − 0.192·27-s + 1.05·28-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: \(0\)
Selberg data: \((2,\ 2013,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.418312628\)
\(L(\frac12)\) \(\approx\) \(1.418312628\)
\(L(\frac{5}{2})\) 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 \( 1 + 0.789T + 8T^{2} \)
5 \( 1 - 13.9T + 125T^{2} \)
7 \( 1 + 21.1T + 343T^{2} \)
13 \( 1 - 41.9T + 2.19e3T^{2} \)
17 \( 1 - 53.5T + 4.91e3T^{2} \)
19 \( 1 - 133.T + 6.85e3T^{2} \)
23 \( 1 + 11.7T + 1.21e4T^{2} \)
29 \( 1 - 42.0T + 2.43e4T^{2} \)
31 \( 1 - 184.T + 2.97e4T^{2} \)
37 \( 1 + 330.T + 5.06e4T^{2} \)
41 \( 1 + 449.T + 6.89e4T^{2} \)
43 \( 1 - 384.T + 7.95e4T^{2} \)
47 \( 1 - 324.T + 1.03e5T^{2} \)
53 \( 1 - 406.T + 1.48e5T^{2} \)
59 \( 1 + 528.T + 2.05e5T^{2} \)
67 \( 1 + 455.T + 3.00e5T^{2} \)
71 \( 1 - 991.T + 3.57e5T^{2} \)
73 \( 1 + 574.T + 3.89e5T^{2} \)
79 \( 1 + 317.T + 4.93e5T^{2} \)
83 \( 1 - 1.49e3T + 5.71e5T^{2} \)
89 \( 1 - 346.T + 7.04e5T^{2} \)
97 \( 1 + 317.T + 9.12e5T^{2} \)
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

−9.107039368300916823706249622936, −8.126827336939912888510270666941, −7.14601988983073029867401085291, −6.24047085761404277461686341339, −5.61214900770053954254397189291, −5.06107498603109171277791519055, −3.79281946255061921592530060211, −2.98149239752710175370080975960, −1.48379818994082871827719105482, −0.64480620402963660123922626835, 0.64480620402963660123922626835, 1.48379818994082871827719105482, 2.98149239752710175370080975960, 3.79281946255061921592530060211, 5.06107498603109171277791519055, 5.61214900770053954254397189291, 6.24047085761404277461686341339, 7.14601988983073029867401085291, 8.126827336939912888510270666941, 9.107039368300916823706249622936

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