Properties

Label 2-2013-1.1-c3-0-229
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.476·2-s + 3·3-s − 7.77·4-s + 17.0·5-s − 1.43·6-s − 27.8·7-s + 7.52·8-s + 9·9-s − 8.12·10-s − 11·11-s − 23.3·12-s + 32.3·13-s + 13.2·14-s + 51.1·15-s + 58.5·16-s − 67.3·17-s − 4.29·18-s + 107.·19-s − 132.·20-s − 83.4·21-s + 5.24·22-s − 79.8·23-s + 22.5·24-s + 165.·25-s − 15.4·26-s + 27·27-s + 216.·28-s + ⋯
L(s)  = 1  − 0.168·2-s + 0.577·3-s − 0.971·4-s + 1.52·5-s − 0.0973·6-s − 1.50·7-s + 0.332·8-s + 0.333·9-s − 0.256·10-s − 0.301·11-s − 0.560·12-s + 0.689·13-s + 0.253·14-s + 0.879·15-s + 0.915·16-s − 0.960·17-s − 0.0562·18-s + 1.29·19-s − 1.48·20-s − 0.866·21-s + 0.0508·22-s − 0.723·23-s + 0.191·24-s + 1.32·25-s − 0.116·26-s + 0.192·27-s + 1.45·28-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: \(1\)
Selberg data: \((2,\ 2013,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 0.476T + 8T^{2} \)
5 \( 1 - 17.0T + 125T^{2} \)
7 \( 1 + 27.8T + 343T^{2} \)
13 \( 1 - 32.3T + 2.19e3T^{2} \)
17 \( 1 + 67.3T + 4.91e3T^{2} \)
19 \( 1 - 107.T + 6.85e3T^{2} \)
23 \( 1 + 79.8T + 1.21e4T^{2} \)
29 \( 1 + 27.8T + 2.43e4T^{2} \)
31 \( 1 + 328.T + 2.97e4T^{2} \)
37 \( 1 - 119.T + 5.06e4T^{2} \)
41 \( 1 - 73.9T + 6.89e4T^{2} \)
43 \( 1 - 382.T + 7.95e4T^{2} \)
47 \( 1 + 144.T + 1.03e5T^{2} \)
53 \( 1 + 269.T + 1.48e5T^{2} \)
59 \( 1 - 618.T + 2.05e5T^{2} \)
67 \( 1 + 577.T + 3.00e5T^{2} \)
71 \( 1 - 265.T + 3.57e5T^{2} \)
73 \( 1 - 805.T + 3.89e5T^{2} \)
79 \( 1 + 940.T + 4.93e5T^{2} \)
83 \( 1 + 1.13e3T + 5.71e5T^{2} \)
89 \( 1 + 1.04e3T + 7.04e5T^{2} \)
97 \( 1 - 1.72e3T + 9.12e5T^{2} \)
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

−8.799088134994071790982740711543, −7.70213284734941395996731595369, −6.78110350529806184072952302084, −5.89515373704576002244865266302, −5.42214484723679331344623848665, −4.14803271888916308179694144893, −3.33572034009921518671662384088, −2.40764666794402133491279627940, −1.29016524650164218258560436012, 0, 1.29016524650164218258560436012, 2.40764666794402133491279627940, 3.33572034009921518671662384088, 4.14803271888916308179694144893, 5.42214484723679331344623848665, 5.89515373704576002244865266302, 6.78110350529806184072952302084, 7.70213284734941395996731595369, 8.799088134994071790982740711543

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