↑

Properties

Label 2-2646-63.4-c1-0-20
Degree $2$
Conductor $2646$
Sign $0.975 + 0.220i$
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 − 2·5-s + 0.999·8-s + (1 − 1.73i)10-s − 11-s + (3 − 5.19i)13-s + (−0.5 + 0.866i)16-s + (−2.5 + 4.33i)17-s + (3.5 + 6.06i)19-s + (0.999 + 1.73i)20-s + (0.5 − 0.866i)22-s − 4·23-s − 25-s + (3 + 5.19i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.894·5-s + 0.353·8-s + (0.316 − 0.547i)10-s − 0.301·11-s + (0.832 − 1.44i)13-s + (−0.125 + 0.216i)16-s + (−0.606 + 1.05i)17-s + (0.802 + 1.39i)19-s + (0.223 + 0.387i)20-s + (0.106 − 0.184i)22-s − 0.834·23-s − 0.200·25-s + (0.588 + 1.01i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.975 + 0.220i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.975 + 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9313027608\)
\(L(\frac12)\) \(\approx\) \(0.9313027608\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T + 5T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.5 + 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8 - 13.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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

−8.409856516810259143372194725757, −8.074185992594133302064861435516, −7.64026422292377683334903142510, −6.49628179168409664082774137112, −5.85529277817401938098218486611, −5.10993824603754659375049173356, −3.89810385030482013172080098630, −3.44865069292980736599652054521, −1.86448457309065429023780747652, −0.48561583303596323392159799103, 0.838039241967278125687629409221, 2.18582867691027681027792759570, 3.09940198233352821221106510789, 4.14951385249271210161872419794, 4.56624098498399379404468894281, 5.77643839851255798247956702338, 6.91278515068091232076092668255, 7.37361004215331648845711461191, 8.254029065573014026360465971436, 9.051773431572888299287554977646

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