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Properties

Label 2-288-72.13-c1-0-4
Degree $2$
Conductor $288$
Sign $0.900 - 0.435i$
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  + (1.52 + 0.816i)3-s + (0.602 + 0.348i)5-s + (−0.795 − 1.37i)7-s + (1.66 + 2.49i)9-s + (2.37 − 1.36i)11-s + (4.76 + 2.75i)13-s + (0.636 + 1.02i)15-s − 5.65·17-s − 0.963i·19-s + (−0.0906 − 2.75i)21-s + (−3.28 + 5.69i)23-s + (−2.25 − 3.91i)25-s + (0.512 + 5.17i)27-s + (−2.85 + 1.64i)29-s + (3.69 − 6.40i)31-s + ⋯
L(s)  = 1  + (0.882 + 0.471i)3-s + (0.269 + 0.155i)5-s + (−0.300 − 0.520i)7-s + (0.555 + 0.831i)9-s + (0.715 − 0.412i)11-s + (1.32 + 0.763i)13-s + (0.164 + 0.264i)15-s − 1.37·17-s − 0.221i·19-s + (−0.0197 − 0.600i)21-s + (−0.685 + 1.18i)23-s + (−0.451 − 0.782i)25-s + (0.0985 + 0.995i)27-s + (−0.529 + 0.305i)29-s + (0.664 − 1.15i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.900 - 0.435i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.900 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71657 + 0.393368i\)
\(L(\frac12)\) \(\approx\) \(1.71657 + 0.393368i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 - 0.816i)T \)
good5 \( 1 + (-0.602 - 0.348i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.795 + 1.37i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.37 + 1.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.76 - 2.75i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.65T + 17T^{2} \)
19 \( 1 + 0.963iT - 19T^{2} \)
23 \( 1 + (3.28 - 5.69i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.85 - 1.64i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.69 + 6.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.25iT - 37T^{2} \)
41 \( 1 + (0.931 - 1.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.99 - 1.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.85 + 6.67i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.54iT - 53T^{2} \)
59 \( 1 + (4.62 + 2.66i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.93 - 4.58i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.95 - 3.43i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.68T + 71T^{2} \)
73 \( 1 - 2.83T + 73T^{2} \)
79 \( 1 + (2.87 + 4.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.74 - 3.31i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.98T + 89T^{2} \)
97 \( 1 + (1.24 + 2.16i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.63084407971914525833258104658, −10.90541359020385244661101751199, −9.827788213553566333397571714683, −9.080143821000551139645767793748, −8.261362002205426001638895514808, −6.99330393227540091917565938919, −6.03280066101890343184430374984, −4.29126776342993869601597105745, −3.59283624698319937790697761078, −1.94554931999248493080599754496, 1.65975776037778050539330345158, 3.06765911372400327244720409331, 4.31976710521755479255577600596, 6.03749935372696757312346598744, 6.75339341008954056515620770648, 8.133030506771170934548903838461, 8.810810696425644294789322788844, 9.605225407462879370046996412325, 10.75535355352678919514445844438, 11.97049105698892480377628238560

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