Properties

Label 2-24e2-8.5-c3-0-15
Degree $2$
Conductor $576$
Sign $0.707 + 0.707i$
Analytic cond. $33.9851$
Root an. cond. $5.82967$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 31.1·7-s + 62.3i·13-s − 56i·19-s + 125·25-s + 155.·31-s − 436. i·37-s − 520i·43-s + 629·49-s + 935. i·61-s − 880i·67-s + 1.19e3·73-s − 1.09e3·79-s − 1.94e3i·91-s + 1.33e3·97-s + 1.02e3·103-s + ⋯
L(s)  = 1  − 1.68·7-s + 1.33i·13-s − 0.676i·19-s + 25-s + 0.903·31-s − 1.93i·37-s − 1.84i·43-s + 1.83·49-s + 1.96i·61-s − 1.60i·67-s + 1.90·73-s − 1.55·79-s − 2.23i·91-s + 1.39·97-s + 0.984·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(33.9851\)
Root analytic conductor: \(5.82967\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :3/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.230239455\)
\(L(\frac12)\) \(\approx\) \(1.230239455\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 125T^{2} \)
7 \( 1 + 31.1T + 343T^{2} \)
11 \( 1 - 1.33e3T^{2} \)
13 \( 1 - 62.3iT - 2.19e3T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 + 56iT - 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 - 155.T + 2.97e4T^{2} \)
37 \( 1 + 436. iT - 5.06e4T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 + 520iT - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 - 2.05e5T^{2} \)
61 \( 1 - 935. iT - 2.26e5T^{2} \)
67 \( 1 + 880iT - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 1.19e3T + 3.89e5T^{2} \)
79 \( 1 + 1.09e3T + 4.93e5T^{2} \)
83 \( 1 - 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 - 1.33e3T + 9.12e5T^{2} \)
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

−10.12584164306680986796151648132, −9.261783059548057623196480300397, −8.776322269845602539813171521013, −7.20299662392624029509997116764, −6.70807045534843305589730795178, −5.76527379912107307897262210975, −4.43037369039784238661009507917, −3.41849416572924098496586837432, −2.30206922030250597574865332382, −0.49048782530309787635686146665, 0.871785814410549917259442415044, 2.81779764860771217270719786386, 3.44385088941095207008093196323, 4.87141050568116483246671147657, 6.05632386883156951524886598175, 6.63281908547253912362658800859, 7.80090089332667601244826450045, 8.668542282174829106575956717673, 9.892963060928200102167981032395, 10.06913628748545919325503774285

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