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Properties

Label 2-288-32.29-c1-0-10
Degree $2$
Conductor $288$
Sign $0.137 + 0.990i$
Analytic cond. $2.29969$
Root an. cond. $1.51647$
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  + (−0.525 − 1.31i)2-s + (−1.44 + 1.38i)4-s + (0.155 + 0.375i)5-s + (0.709 − 0.709i)7-s + (2.57 + 1.17i)8-s + (0.411 − 0.401i)10-s + (2.79 − 1.15i)11-s + (2.58 − 6.24i)13-s + (−1.30 − 0.558i)14-s + (0.187 − 3.99i)16-s − 1.05i·17-s + (−1.48 + 3.59i)19-s + (−0.743 − 0.328i)20-s + (−2.98 − 3.05i)22-s + (−0.922 − 0.922i)23-s + ⋯
L(s)  = 1  + (−0.371 − 0.928i)2-s + (−0.723 + 0.690i)4-s + (0.0696 + 0.168i)5-s + (0.268 − 0.268i)7-s + (0.909 + 0.414i)8-s + (0.130 − 0.127i)10-s + (0.841 − 0.348i)11-s + (0.717 − 1.73i)13-s + (−0.348 − 0.149i)14-s + (0.0468 − 0.998i)16-s − 0.256i·17-s + (−0.341 + 0.823i)19-s + (−0.166 − 0.0735i)20-s + (−0.636 − 0.651i)22-s + (−0.192 − 0.192i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.137 + 0.990i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.137 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.817688 - 0.712105i\)
\(L(\frac12)\) \(\approx\) \(0.817688 - 0.712105i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.525 + 1.31i)T \)
3 \( 1 \)
good5 \( 1 + (-0.155 - 0.375i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-0.709 + 0.709i)T - 7iT^{2} \)
11 \( 1 + (-2.79 + 1.15i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-2.58 + 6.24i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 1.05iT - 17T^{2} \)
19 \( 1 + (1.48 - 3.59i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.922 + 0.922i)T + 23iT^{2} \)
29 \( 1 + (-7.64 - 3.16i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 1.88T + 31T^{2} \)
37 \( 1 + (-1.24 - 3.01i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (5.11 + 5.11i)T + 41iT^{2} \)
43 \( 1 + (10.9 - 4.53i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 7.47iT - 47T^{2} \)
53 \( 1 + (-7.58 + 3.14i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-4.13 - 9.98i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-1.35 - 0.562i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-10.8 - 4.50i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (9.35 - 9.35i)T - 71iT^{2} \)
73 \( 1 + (-0.367 - 0.367i)T + 73iT^{2} \)
79 \( 1 + 5.87iT - 79T^{2} \)
83 \( 1 + (1.62 - 3.91i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (8.33 - 8.33i)T - 89iT^{2} \)
97 \( 1 + 7.97T + 97T^{2} \)
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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

−11.53981645929571204910018956087, −10.48685208207098130576467080545, −10.12026717694368779712745623045, −8.616100437041779050965002115388, −8.209240265035921087610485241968, −6.78092872469091061119898420154, −5.36940441870057371129474339851, −3.98198026765945709617204407220, −2.92891673397608916563535993476, −1.12198907155525564990396657597, 1.62426343524441129626639744448, 4.05459509309118105902051783748, 5.01030554422727867451481718239, 6.41796790763096143146229691817, 6.91650591226229420507232379695, 8.363452012347883790164011177612, 8.992117566939897894477472741433, 9.791806424683655487233170440456, 11.07203120879373003558857515684, 11.91853017951933806123124591887

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