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Properties

Label 2-288-36.11-c1-0-11
Degree $2$
Conductor $288$
Sign $-0.726 + 0.687i$
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  + (0.324 − 1.70i)3-s + (−1.68 + 0.971i)5-s + (−2.61 − 1.50i)7-s + (−2.78 − 1.10i)9-s + (2.20 − 3.81i)11-s + (−2.65 − 4.60i)13-s + (1.10 + 3.17i)15-s + 4.16i·17-s − 4.66i·19-s + (−3.41 + 3.95i)21-s + (1.30 + 2.26i)23-s + (−0.613 + 1.06i)25-s + (−2.78 + 4.38i)27-s + (1.13 + 0.655i)29-s + (−0.0648 + 0.0374i)31-s + ⋯
L(s)  = 1  + (0.187 − 0.982i)3-s + (−0.752 + 0.434i)5-s + (−0.988 − 0.570i)7-s + (−0.929 − 0.367i)9-s + (0.664 − 1.15i)11-s + (−0.737 − 1.27i)13-s + (0.285 + 0.820i)15-s + 1.00i·17-s − 1.06i·19-s + (−0.745 + 0.863i)21-s + (0.272 + 0.471i)23-s + (−0.122 + 0.212i)25-s + (−0.535 + 0.844i)27-s + (0.210 + 0.121i)29-s + (−0.0116 + 0.00672i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.726 + 0.687i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.726 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.300904 - 0.755699i\)
\(L(\frac12)\) \(\approx\) \(0.300904 - 0.755699i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.324 + 1.70i)T \)
good5 \( 1 + (1.68 - 0.971i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.61 + 1.50i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.20 + 3.81i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.65 + 4.60i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.16iT - 17T^{2} \)
19 \( 1 + 4.66iT - 19T^{2} \)
23 \( 1 + (-1.30 - 2.26i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.13 - 0.655i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.0648 - 0.0374i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.43T + 37T^{2} \)
41 \( 1 + (-8.16 + 4.71i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.06 + 2.92i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.40 + 4.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.96iT - 53T^{2} \)
59 \( 1 + (1.74 + 3.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.32 + 10.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.38 + 1.37i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.910T + 71T^{2} \)
73 \( 1 - 6.86T + 73T^{2} \)
79 \( 1 + (10.8 + 6.28i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.61 - 14.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.96iT - 89T^{2} \)
97 \( 1 + (9.03 - 15.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.45414735770344839467069021075, −10.75725905212435569747155659072, −9.477124422438075391608019344572, −8.358508908447409407075250388989, −7.48968331932132729683160384496, −6.69118004869125473926570152862, −5.71585678953664524274469827529, −3.72246335327397018993065911975, −2.90354510077400119736944054105, −0.58677093631218167684459183012, 2.59715502059394576224588100428, 4.06934018509146197775397427938, 4.70952368568192743707262677088, 6.19121556250956628907375020823, 7.36604985620460546785386800121, 8.601966196392858852941504357510, 9.617584247988943132361948645139, 9.790938735348782682600998491782, 11.45408935577543058850902608977, 12.00582162217602606818390726573

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