↑

Properties

Label 2-513-171.106-c1-0-10
Degree $2$
Conductor $513$
Sign $-0.361 + 0.932i$
Analytic cond. $4.09632$
Root an. cond. $2.02393$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s + 5-s + (−1.5 + 2.59i)7-s − 3·8-s + (−0.5 − 0.866i)10-s + (1.5 − 2.59i)11-s + (3 − 5.19i)13-s + 3·14-s + (0.500 + 0.866i)16-s + (1.5 − 2.59i)17-s + (−4 − 1.73i)19-s + (0.500 − 0.866i)20-s − 3·22-s + (4 − 6.92i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s + 0.447·5-s + (−0.566 + 0.981i)7-s − 1.06·8-s + (−0.158 − 0.273i)10-s + (0.452 − 0.783i)11-s + (0.832 − 1.44i)13-s + 0.801·14-s + (0.125 + 0.216i)16-s + (0.363 − 0.630i)17-s + (−0.917 − 0.397i)19-s + (0.111 − 0.193i)20-s − 0.639·22-s + (0.834 − 1.44i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $-0.361 + 0.932i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ -0.361 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.689256 - 1.00696i\)
\(L(\frac12)\) \(\approx\) \(0.689256 - 1.00696i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (4 + 1.73i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - T + 5T^{2} \)
7 \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - T + 41T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9T + 47T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.5 - 12.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.53982779185925441555393422261, −9.914733318800735856680820690356, −8.849058922344088454146548866671, −8.443471412362247655743573755376, −6.59846699055009858271107852262, −6.06570041081805960450976585646, −5.17165467997458954996451952542, −3.24570181529285020852700786694, −2.50510013608551059842848175087, −0.837857107130942695788145712451, 1.75576313996937568804761698360, 3.47959918976036542031719361535, 4.33990420847873951360144415563, 6.13226787349324150529786119100, 6.58985942488832300076487238506, 7.45918675433093412309689321841, 8.396419621598843879061991890148, 9.422274484681160721014995885668, 9.977551879038831302476544619924, 11.22764068267892271213048943580

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