↑

Properties

Label 2-2268-252.83-c0-0-8
Degree $2$
Conductor $2268$
Sign $0.342 + 0.939i$
Analytic cond. $1.13187$
Root an. cond. $1.06389$
Motivic weight $0$
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 + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s − 0.999·8-s + 0.999·10-s + (−0.5 + 0.866i)11-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 2·17-s + 19-s + (0.499 − 0.866i)20-s + (0.499 + 0.866i)22-s + (−0.5 − 0.866i)23-s − 0.999·28-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s − 0.999·8-s + 0.999·10-s + (−0.5 + 0.866i)11-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 2·17-s + 19-s + (0.499 − 0.866i)20-s + (0.499 + 0.866i)22-s + (−0.5 − 0.866i)23-s − 0.999·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ 0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.671813220\)
\(L(\frac12)\) \(\approx\) \(1.671813220\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - 2T + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−9.633467101347565511154702836330, −8.196326751296965268077681621857, −7.49958351367229377470887375163, −6.67632847316378803248875895481, −5.68826070690439759320572913694, −5.02979612245365128530448139990, −4.06108478928931631392451988095, −3.20879248805190118779231409645, −2.32213462980658126936753311184, −1.23122936992700661333303101077, 1.39950496191664247000278156541, 2.92511609828972627507093950251, 3.69450541689916265986934381579, 5.07041090306899905344249599317, 5.51453117300928735154655764194, 5.74277004547041211131294746063, 7.09217965295457934381414450408, 7.925445301327123315978242036883, 8.386964245086595000936451757591, 9.206498672716002321955975821388

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