↑

Properties

Label 2-48e2-36.7-c0-0-0
Degree $2$
Conductor $2304$
Sign $0.642 - 0.766i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
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  + (0.866 + 0.5i)3-s + (0.499 + 0.866i)9-s + (0.866 + 0.5i)11-s − 17-s + i·19-s + (0.5 − 0.866i)25-s + 0.999i·27-s + (0.499 + 0.866i)33-s + (−0.5 − 0.866i)41-s + (0.866 + 0.5i)43-s + (−0.5 − 0.866i)49-s + (−0.866 − 0.5i)51-s + (−0.5 + 0.866i)57-s + (−0.866 + 0.5i)59-s + (0.866 − 0.5i)67-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.499 + 0.866i)9-s + (0.866 + 0.5i)11-s − 17-s + i·19-s + (0.5 − 0.866i)25-s + 0.999i·27-s + (0.499 + 0.866i)33-s + (−0.5 − 0.866i)41-s + (0.866 + 0.5i)43-s + (−0.5 − 0.866i)49-s + (−0.866 − 0.5i)51-s + (−0.5 + 0.866i)57-s + (−0.866 + 0.5i)59-s + (0.866 − 0.5i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ 0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.649686820\)
\(L(\frac12)\) \(\approx\) \(1.649686820\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.266879418812358714571239849377, −8.600272022583080256861203043462, −7.938890457744806662446581020588, −7.02557030847517048691633401388, −6.32502592449527237233284336875, −5.15308494852303517004346873195, −4.29319001671565954305860173760, −3.71250563335131941767922600289, −2.57929052012426391999492124663, −1.66138205462997518508901802122, 1.15721029058040105332646797819, 2.31656697144977186146516788183, 3.22886789385100624913097128544, 4.07922018546408368656316682214, 5.02403469423489059128847883666, 6.30507226751181407969253879913, 6.74909087841298304548721457570, 7.57803823632743325442480845088, 8.389004882911866079291867321960, 9.178280169884842177503465970088

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