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Properties

Label 2-24e2-8.5-c7-0-31
Degree $2$
Conductor $576$
Sign $0.965 - 0.258i$
Analytic cond. $179.933$
Root an. cond. $13.4139$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 446. i·5-s − 1.63e3·7-s + 7.53e3i·11-s − 5.16e3i·13-s − 2.93e3·17-s − 4.69e4i·19-s + 5.06e4·23-s − 1.20e5·25-s + 1.64e4i·29-s − 4.18e4·31-s − 7.28e5i·35-s − 3.42e5i·37-s − 6.42e5·41-s − 5.72e5i·43-s − 8.04e5·47-s + ⋯
L(s)  = 1  + 1.59i·5-s − 1.79·7-s + 1.70i·11-s − 0.652i·13-s − 0.144·17-s − 1.57i·19-s + 0.867·23-s − 1.54·25-s + 0.125i·29-s − 0.252·31-s − 2.87i·35-s − 1.11i·37-s − 1.45·41-s − 1.09i·43-s − 1.12·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.965 - 0.258i$
Analytic conductor: \(179.933\)
Root analytic conductor: \(13.4139\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :7/2),\ 0.965 - 0.258i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.9580914228\)
\(L(\frac12)\) \(\approx\) \(0.9580914228\)
\(L(\frac{9}{2})\) 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 \( 1 - 446. iT - 7.81e4T^{2} \)
7 \( 1 + 1.63e3T + 8.23e5T^{2} \)
11 \( 1 - 7.53e3iT - 1.94e7T^{2} \)
13 \( 1 + 5.16e3iT - 6.27e7T^{2} \)
17 \( 1 + 2.93e3T + 4.10e8T^{2} \)
19 \( 1 + 4.69e4iT - 8.93e8T^{2} \)
23 \( 1 - 5.06e4T + 3.40e9T^{2} \)
29 \( 1 - 1.64e4iT - 1.72e10T^{2} \)
31 \( 1 + 4.18e4T + 2.75e10T^{2} \)
37 \( 1 + 3.42e5iT - 9.49e10T^{2} \)
41 \( 1 + 6.42e5T + 1.94e11T^{2} \)
43 \( 1 + 5.72e5iT - 2.71e11T^{2} \)
47 \( 1 + 8.04e5T + 5.06e11T^{2} \)
53 \( 1 - 5.29e5iT - 1.17e12T^{2} \)
59 \( 1 + 1.71e6iT - 2.48e12T^{2} \)
61 \( 1 - 2.91e6iT - 3.14e12T^{2} \)
67 \( 1 + 2.41e6iT - 6.06e12T^{2} \)
71 \( 1 - 2.16e6T + 9.09e12T^{2} \)
73 \( 1 - 2.11e6T + 1.10e13T^{2} \)
79 \( 1 - 2.20e5T + 1.92e13T^{2} \)
83 \( 1 - 3.13e6iT - 2.71e13T^{2} \)
89 \( 1 + 3.97e6T + 4.42e13T^{2} \)
97 \( 1 - 1.67e7T + 8.07e13T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.795881441606983480849002021945, −9.043874627505141759068354082672, −7.38167913857252705056254267779, −6.94102550536165924344131666212, −6.40516632245378535893503191889, −5.09414523555985356838459998405, −3.71034157863024338512228318088, −2.94482811983970779541168955385, −2.22860468727704385052209238701, −0.31451073938951921209403034546, 0.54311467444249877520827914693, 1.47084024574486420207225345787, 3.12710152046763203217750861144, 3.80463092271853501251068510277, 5.05307727182872227635872364268, 5.97994051812687648146531862091, 6.61022162706883488800800163766, 8.105506618479478937864885332287, 8.738126731781830357852080050240, 9.455303559263564412245605434151

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