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Properties

Label 2-288-32.29-c1-0-16
Degree $2$
Conductor $288$
Sign $-0.679 + 0.733i$
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  + (0.582 − 1.28i)2-s + (−1.32 − 1.50i)4-s + (−0.445 − 1.07i)5-s + (2.57 − 2.57i)7-s + (−2.70 + 0.829i)8-s + (−1.64 − 0.0521i)10-s + (−2.20 + 0.914i)11-s + (0.264 − 0.638i)13-s + (−1.81 − 4.81i)14-s + (−0.505 + 3.96i)16-s − 2.58i·17-s + (−0.695 + 1.67i)19-s + (−1.02 + 2.09i)20-s + (−0.106 + 3.37i)22-s + (−5.69 − 5.69i)23-s + ⋯
L(s)  = 1  + (0.411 − 0.911i)2-s + (−0.660 − 0.750i)4-s + (−0.199 − 0.481i)5-s + (0.973 − 0.973i)7-s + (−0.956 + 0.293i)8-s + (−0.520 − 0.0164i)10-s + (−0.665 + 0.275i)11-s + (0.0733 − 0.177i)13-s + (−0.486 − 1.28i)14-s + (−0.126 + 0.991i)16-s − 0.626i·17-s + (−0.159 + 0.385i)19-s + (−0.229 + 0.467i)20-s + (−0.0228 + 0.719i)22-s + (−1.18 − 1.18i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.679 + 0.733i$
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: \((2,\ 288,\ (\ :1/2),\ -0.679 + 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.567111 - 1.29874i\)
\(L(\frac12)\) \(\approx\) \(0.567111 - 1.29874i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.582 + 1.28i)T \)
3 \( 1 \)
good5 \( 1 + (0.445 + 1.07i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-2.57 + 2.57i)T - 7iT^{2} \)
11 \( 1 + (2.20 - 0.914i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-0.264 + 0.638i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 2.58iT - 17T^{2} \)
19 \( 1 + (0.695 - 1.67i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (5.69 + 5.69i)T + 23iT^{2} \)
29 \( 1 + (-7.43 - 3.07i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 6.28T + 31T^{2} \)
37 \( 1 + (-2.64 - 6.37i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-7.15 - 7.15i)T + 41iT^{2} \)
43 \( 1 + (1.64 - 0.681i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 7.69iT - 47T^{2} \)
53 \( 1 + (-5.90 + 2.44i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (1.43 + 3.47i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-8.51 - 3.52i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (8.92 + 3.69i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (2.02 - 2.02i)T - 71iT^{2} \)
73 \( 1 + (10.7 + 10.7i)T + 73iT^{2} \)
79 \( 1 - 0.523iT - 79T^{2} \)
83 \( 1 + (4.73 - 11.4i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-9.44 + 9.44i)T - 89iT^{2} \)
97 \( 1 + 4.39T + 97T^{2} \)
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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

−11.54520429258729116408396489872, −10.50012040032001181839329033544, −10.04485764748582549181340376520, −8.563312080357444463888710973825, −7.87070605535392467176862727265, −6.30916427872838444612131147804, −4.76194605068771651066074520266, −4.42817331253519942756834702676, −2.71123061938271715878048781220, −1.02912358600502659235084555587, 2.57233674638563629995652627462, 4.08252177573562066114620598256, 5.29749309353693470149726127317, 6.07610222223271443176540661588, 7.35097665324587421873277876327, 8.203226338568419869155317819869, 8.907743456277735835590547557024, 10.27574389962606264470141193547, 11.49685131245063634502652820541, 12.12826911965312477596450435330

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