Properties

Label 2-24e2-8.5-c3-0-13
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

\(L(2)\) \(\approx\) \(2.376640124\)
\(L(\frac12)\) \(\approx\) \(2.376640124\)
\(L(\frac{5}{2})\) 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

−10.67323897066444768780772321516, −9.409620230906231153855156709231, −8.659519045735364960050246875752, −7.78911455045350943659198408269, −6.96998242271999813662195366304, −5.72967424496039947151949618367, −4.76443069749800868349116880033, −3.95357947718611103681208369815, −2.24831316230899120951329007367, −1.27866218438550636600268532293, 0.794340645454267792128082113077, 2.07427991504094383689732004954, 3.41093966499436541116066998530, 4.85642264996169452708530555221, 5.27128099269900918543880838778, 6.65586898992746637698457271351, 7.76331001373454494378940423164, 8.280421395757425529257094247782, 9.219628665599115125501251980461, 10.49912455618165190277473469522

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