↑

Properties

Label 2-2646-63.16-c1-0-33
Degree $2$
Conductor $2646$
Sign $-0.959 + 0.282i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
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 + (−0.499 + 0.866i)4-s − 0.517·5-s + 0.999·8-s + (0.258 + 0.448i)10-s + 1.46·11-s + (1.22 + 2.12i)13-s + (−0.5 − 0.866i)16-s + (−1.74 − 3.01i)17-s + (−0.258 + 0.448i)19-s + (0.258 − 0.448i)20-s + (−0.732 − 1.26i)22-s − 7.92·23-s − 4.73·25-s + (1.22 − 2.12i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.231·5-s + 0.353·8-s + (0.0818 + 0.141i)10-s + 0.441·11-s + (0.339 + 0.588i)13-s + (−0.125 − 0.216i)16-s + (−0.422 − 0.731i)17-s + (−0.0593 + 0.102i)19-s + (0.0578 − 0.100i)20-s + (−0.156 − 0.270i)22-s − 1.65·23-s − 0.946·25-s + (0.240 − 0.416i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.959 + 0.282i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.959 + 0.282i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5613371260\)
\(L(\frac12)\) \(\approx\) \(0.5613371260\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 0.517T + 5T^{2} \)
11 \( 1 - 1.46T + 11T^{2} \)
13 \( 1 + (-1.22 - 2.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.74 + 3.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.258 - 0.448i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.92T + 23T^{2} \)
29 \( 1 + (-1.36 + 2.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.67 - 6.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.82 - 4.89i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.09 + 10.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.31 + 4.00i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.36 - 5.83i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.39 + 12.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.19 + 3.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.90 + 3.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.803T + 71T^{2} \)
73 \( 1 + (2.31 + 4.00i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.06 + 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.94 - 8.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.05 - 13.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.517 + 0.896i)T + (-48.5 - 84.0i)T^{2} \)
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

−8.619854584453875773669288528656, −7.888846597774580173378807350242, −7.11654391170977658116426238405, −6.27484518201592522622357314533, −5.35056130533735943602460255666, −4.16435752283130689181889593133, −3.80423722448343225007991847181, −2.51350481575236235270071782156, −1.66450807589180568164678662977, −0.21645570023597463384106240509, 1.27045115915773789580008092107, 2.49315691667108990187887717681, 3.87932181298461306180686263690, 4.35216390635075693218450542485, 5.71152087237810480092399416772, 6.03522775375318443505261177831, 6.96414849926735747411425547893, 7.86789812351547456763733570470, 8.239760211525327233311803555403, 9.107155217165369714781745772483

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