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Properties

Label 2-288-96.35-c1-0-9
Degree $2$
Conductor $288$
Sign $0.0978 + 0.995i$
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  + (−0.349 − 1.37i)2-s + (−1.75 + 0.958i)4-s + (2.97 − 1.23i)5-s + (0.237 + 0.237i)7-s + (1.92 + 2.06i)8-s + (−2.72 − 3.64i)10-s + (2.12 − 0.879i)11-s + (0.0390 − 0.0943i)13-s + (0.242 − 0.408i)14-s + (2.16 − 3.36i)16-s + 4.16·17-s + (−4.25 − 1.76i)19-s + (−4.03 + 5.01i)20-s + (−1.94 − 2.60i)22-s + (−4.84 − 4.84i)23-s + ⋯
L(s)  = 1  + (−0.247 − 0.968i)2-s + (−0.877 + 0.479i)4-s + (1.33 − 0.550i)5-s + (0.0898 + 0.0898i)7-s + (0.681 + 0.731i)8-s + (−0.862 − 1.15i)10-s + (0.640 − 0.265i)11-s + (0.0108 − 0.0261i)13-s + (0.0648 − 0.109i)14-s + (0.540 − 0.841i)16-s + 1.00·17-s + (−0.975 − 0.404i)19-s + (−0.903 + 1.12i)20-s + (−0.415 − 0.554i)22-s + (−1.01 − 1.01i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0978 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0978 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.0978 + 0.995i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.0978 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.977570 - 0.886165i\)
\(L(\frac12)\) \(\approx\) \(0.977570 - 0.886165i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.349 + 1.37i)T \)
3 \( 1 \)
good5 \( 1 + (-2.97 + 1.23i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-0.237 - 0.237i)T + 7iT^{2} \)
11 \( 1 + (-2.12 + 0.879i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-0.0390 + 0.0943i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 4.16T + 17T^{2} \)
19 \( 1 + (4.25 + 1.76i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (4.84 + 4.84i)T + 23iT^{2} \)
29 \( 1 + (-2.90 + 7.01i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 9.88iT - 31T^{2} \)
37 \( 1 + (-0.175 - 0.424i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (7.67 - 7.67i)T - 41iT^{2} \)
43 \( 1 + (-2.99 - 7.23i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 6.10iT - 47T^{2} \)
53 \( 1 + (-4.28 - 10.3i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-1.19 - 2.88i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-6.29 - 2.60i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (5.66 - 13.6i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-3.49 + 3.49i)T - 71iT^{2} \)
73 \( 1 + (1.42 + 1.42i)T + 73iT^{2} \)
79 \( 1 - 1.53T + 79T^{2} \)
83 \( 1 + (2.95 - 7.14i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (1.92 + 1.92i)T + 89iT^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−11.69552779493077947319868445340, −10.37426956349997750224987151412, −9.928045344553232553848415483150, −8.888615065224435239512823664958, −8.242176314193861662638416182814, −6.48532642121111877766036312925, −5.37832217442254297332797364864, −4.22977161123236885952103368023, −2.62472069315280582991825790006, −1.34319579424866338421062943692, 1.80208104291967349733342060452, 3.85287255865810550930205597318, 5.38904073894184230637787708111, 6.13039281262540193763344772104, 7.00776448404328485569068847878, 8.107079139163255581316984608424, 9.285426775324510629829372176287, 9.929607409310533074466200404370, 10.68582275665338999821415134629, 12.18416714141591710018291738478

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