↑

Properties

Label 2-2646-63.16-c1-0-34
Degree $2$
Conductor $2646$
Sign $-0.920 + 0.389i$
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 − 0.613·5-s + 0.999·8-s + (0.306 + 0.531i)10-s + 4.62·11-s + (−3.25 − 5.64i)13-s + (−0.5 − 0.866i)16-s + (−1.01 − 1.75i)17-s + (−1.32 + 2.29i)19-s + (0.306 − 0.531i)20-s + (−2.31 − 4.00i)22-s + 3.62·23-s − 4.62·25-s + (−3.25 + 5.64i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.274·5-s + 0.353·8-s + (0.0970 + 0.168i)10-s + 1.39·11-s + (−0.903 − 1.56i)13-s + (−0.125 − 0.216i)16-s + (−0.246 − 0.426i)17-s + (−0.303 + 0.525i)19-s + (0.0686 − 0.118i)20-s + (−0.492 − 0.853i)22-s + 0.755·23-s − 0.924·25-s + (−0.639 + 1.10i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.920 + 0.389i$
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: \((2,\ 2646,\ (\ :1/2),\ -0.920 + 0.389i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8222570119\)
\(L(\frac12)\) \(\approx\) \(0.8222570119\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.613T + 5T^{2} \)
11 \( 1 - 4.62T + 11T^{2} \)
13 \( 1 + (3.25 + 5.64i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.01 + 1.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.32 - 2.29i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.62T + 23T^{2} \)
29 \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.25 + 5.64i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.01 + 1.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.31 + 2.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.29 - 9.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.11 + 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.56 + 6.17i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.31 - 4.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.376T + 71T^{2} \)
73 \( 1 + (3.66 + 6.34i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.81 + 11.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.87 + 6.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.13 - 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.14 + 14.1i)T + (-48.5 - 84.0i)T^{2} \)
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

−8.589622843330560933273923873357, −7.85766263107735574076300606054, −7.20898901373187540481559742402, −6.27009053381813652635886279020, −5.30848167403630920331073682208, −4.39626212216822932794864707308, −3.54253562614084696079683756083, −2.73449695963466005612677415678, −1.54504677672543088888453076851, −0.32287934620644146823539808845, 1.31845089179623507615763830680, 2.38996781275699422646323398663, 3.94299562826767483325292920879, 4.35434765886607821728254415691, 5.39371816366015718141081098227, 6.41851203804084664633309364863, 6.90423820693429176023656243970, 7.49437852934185824740576449427, 8.628418427903458525027696244892, 9.037529978127537826183935460784

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