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Properties

Label 2-24e2-72.13-c1-0-9
Degree $2$
Conductor $576$
Sign $0.581 + 0.813i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.57 − 0.724i)3-s + (1.94 + 2.28i)9-s + (−0.476 + 0.275i)11-s + 7.89·17-s − 6.34i·19-s + (−2.5 − 4.33i)25-s + (−1.41 − 5.00i)27-s + (0.949 − 0.0874i)33-s + (3.39 − 5.88i)41-s + (10.6 − 6.17i)43-s + (3.5 − 6.06i)49-s + (−12.4 − 5.72i)51-s + (−4.60 + 9.98i)57-s + (13.2 + 7.62i)59-s + (−0.301 − 0.174i)67-s + ⋯
L(s)  = 1  + (−0.908 − 0.418i)3-s + (0.649 + 0.760i)9-s + (−0.143 + 0.0829i)11-s + 1.91·17-s − 1.45i·19-s + (−0.5 − 0.866i)25-s + (−0.272 − 0.962i)27-s + (0.165 − 0.0152i)33-s + (0.530 − 0.919i)41-s + (1.63 − 0.941i)43-s + (0.5 − 0.866i)49-s + (−1.74 − 0.801i)51-s + (−0.609 + 1.32i)57-s + (1.71 + 0.992i)59-s + (−0.0368 − 0.0212i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.581 + 0.813i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.581 + 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.946984 - 0.487428i\)
\(L(\frac12)\) \(\approx\) \(0.946984 - 0.487428i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.57 + 0.724i)T \)
good5 \( 1 + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.476 - 0.275i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 7.89T + 17T^{2} \)
19 \( 1 + 6.34iT - 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-3.39 + 5.88i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.6 + 6.17i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-13.2 - 7.62i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.301 + 0.174i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (15.5 - 9i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 + (4.84 + 8.39i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.57451552438494008863055141547, −9.985864023866402604814713845560, −8.816018692075732644476759811071, −7.66321660509689695294721404045, −7.05836063559488091313195725306, −5.90144436765270804372013753934, −5.23562760360868363589126330285, −4.04842145871993134739729786189, −2.46330175380719052421928941062, −0.814789059261724674545527029267, 1.26716826790490555037854627125, 3.27914413588829060382317656710, 4.28059255699972150945030229302, 5.56212226889069230459793044706, 5.94703205728863340466046860445, 7.30422674775011646147393570442, 8.065571723148301711765686796964, 9.451779169771977970790137673961, 10.00869273971725558853910445298, 10.81783621867344152446950404335

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