↑

Properties

Label 2-2646-63.16-c1-0-15
Degree $2$
Conductor $2646$
Sign $0.795 - 0.606i$
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 − 3.44·5-s − 0.999·8-s + (−1.72 − 2.98i)10-s − 4·11-s + (−2.12 − 3.67i)13-s + (−0.5 − 0.866i)16-s + (−0.707 − 1.22i)17-s + (−3.13 + 5.43i)19-s + (1.72 − 2.98i)20-s + (−2 − 3.46i)22-s + 8.74·23-s + 6.87·25-s + (2.12 − 3.67i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s − 1.54·5-s − 0.353·8-s + (−0.544 − 0.943i)10-s − 1.20·11-s + (−0.588 − 1.01i)13-s + (−0.125 − 0.216i)16-s + (−0.171 − 0.297i)17-s + (−0.719 + 1.24i)19-s + (0.385 − 0.667i)20-s + (−0.426 − 0.738i)22-s + 1.82·23-s + 1.37·25-s + (0.416 − 0.720i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.795 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.795 - 0.606i$
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: \((2,\ 2646,\ (\ :1/2),\ 0.795 - 0.606i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9929855746\)
\(L(\frac12)\) \(\approx\) \(0.9929855746\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.44T + 5T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (2.12 + 3.67i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.13 - 5.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 8.74T + 23T^{2} \)
29 \( 1 + (-0.563 + 0.976i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.73 + 4.74i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.87 - 4.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.82 + 4.89i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.563 - 0.976i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.03 - 3.51i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.30 - 10.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.15 + 7.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.13 - 5.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.43 + 5.95i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.87T + 71T^{2} \)
73 \( 1 + (-2.21 - 3.82i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.936 - 1.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.32 + 2.29i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.53 - 6.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.50 + 13.0i)T + (-48.5 - 84.0i)T^{2} \)
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

−8.406236543510045976504989569783, −8.162602090403223271899363908751, −7.44251569566937468300581869659, −6.88953133056736891055767521326, −5.70753622987016998903998577812, −5.01827023508584100838671072180, −4.28204420940353807982266658894, −3.36966543545860379647076597035, −2.62469079405461084408852419995, −0.53354422669433327057079258081, 0.60327697687399959557287100999, 2.25062201457549848615663319680, 3.07947252815689244431156383709, 3.97207417346207933290442850304, 4.78555662723612279041278670653, 5.20927486644272285890666320035, 6.87305600788805942915051521731, 7.01362339439343046335006040702, 8.186144274870072207683035904744, 8.696370475750500939029743678988

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