Properties

Label 2-2013-1.1-c3-0-52
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  + 2.10·2-s − 3·3-s − 3.56·4-s + 6.12·5-s − 6.31·6-s − 19.2·7-s − 24.3·8-s + 9·9-s + 12.8·10-s + 11·11-s + 10.7·12-s + 47.8·13-s − 40.6·14-s − 18.3·15-s − 22.7·16-s + 35.6·17-s + 18.9·18-s − 138.·19-s − 21.8·20-s + 57.8·21-s + 23.1·22-s + 16.0·23-s + 73.0·24-s − 87.4·25-s + 100.·26-s − 27·27-s + 68.8·28-s + ⋯
L(s)  = 1  + 0.744·2-s − 0.577·3-s − 0.446·4-s + 0.547·5-s − 0.429·6-s − 1.04·7-s − 1.07·8-s + 0.333·9-s + 0.407·10-s + 0.301·11-s + 0.257·12-s + 1.02·13-s − 0.775·14-s − 0.316·15-s − 0.354·16-s + 0.508·17-s + 0.248·18-s − 1.67·19-s − 0.244·20-s + 0.601·21-s + 0.224·22-s + 0.145·23-s + 0.621·24-s − 0.699·25-s + 0.759·26-s − 0.192·27-s + 0.464·28-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

\(L(2)\) \(\approx\) \(1.419583512\)
\(L(\frac12)\) \(\approx\) \(1.419583512\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.10T + 8T^{2} \)
5 \( 1 - 6.12T + 125T^{2} \)
7 \( 1 + 19.2T + 343T^{2} \)
13 \( 1 - 47.8T + 2.19e3T^{2} \)
17 \( 1 - 35.6T + 4.91e3T^{2} \)
19 \( 1 + 138.T + 6.85e3T^{2} \)
23 \( 1 - 16.0T + 1.21e4T^{2} \)
29 \( 1 + 138.T + 2.43e4T^{2} \)
31 \( 1 - 218.T + 2.97e4T^{2} \)
37 \( 1 + 258.T + 5.06e4T^{2} \)
41 \( 1 + 128.T + 6.89e4T^{2} \)
43 \( 1 + 311.T + 7.95e4T^{2} \)
47 \( 1 - 407.T + 1.03e5T^{2} \)
53 \( 1 + 546.T + 1.48e5T^{2} \)
59 \( 1 + 386.T + 2.05e5T^{2} \)
67 \( 1 - 197.T + 3.00e5T^{2} \)
71 \( 1 - 941.T + 3.57e5T^{2} \)
73 \( 1 - 105.T + 3.89e5T^{2} \)
79 \( 1 - 122.T + 4.93e5T^{2} \)
83 \( 1 - 721.T + 5.71e5T^{2} \)
89 \( 1 + 152.T + 7.04e5T^{2} \)
97 \( 1 + 707.T + 9.12e5T^{2} \)
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

−8.900951754867629157732557415497, −8.131935045432382275141756193313, −6.70642729555330081870048122251, −6.26235523309264650212328569595, −5.71024920499428654704679347615, −4.77528640024268021916396985432, −3.86820729059032672869848258485, −3.24669329043654191135561260723, −1.88273265575047182330171583639, −0.49543562071670982845137841051, 0.49543562071670982845137841051, 1.88273265575047182330171583639, 3.24669329043654191135561260723, 3.86820729059032672869848258485, 4.77528640024268021916396985432, 5.71024920499428654704679347615, 6.26235523309264650212328569595, 6.70642729555330081870048122251, 8.131935045432382275141756193313, 8.900951754867629157732557415497

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