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Properties

Label 2-288-32.29-c1-0-2
Degree $2$
Conductor $288$
Sign $0.378 - 0.925i$
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  + (−1.11 − 0.874i)2-s + (0.470 + 1.94i)4-s + (0.707 + 1.70i)5-s + (−0.665 + 0.665i)7-s + (1.17 − 2.57i)8-s + (0.707 − 2.51i)10-s + (−3.69 + 1.52i)11-s + (−1.76 + 4.26i)13-s + (1.32 − 0.157i)14-s + (−3.55 + 1.82i)16-s + 3.61i·17-s + (0.194 − 0.470i)19-s + (−2.98 + 2.17i)20-s + (5.44 + 1.52i)22-s + (1.33 + 1.33i)23-s + ⋯
L(s)  = 1  + (−0.785 − 0.618i)2-s + (0.235 + 0.971i)4-s + (0.316 + 0.763i)5-s + (−0.251 + 0.251i)7-s + (0.416 − 0.909i)8-s + (0.223 − 0.795i)10-s + (−1.11 + 0.461i)11-s + (−0.489 + 1.18i)13-s + (0.353 − 0.0420i)14-s + (−0.889 + 0.457i)16-s + 0.877i·17-s + (0.0446 − 0.107i)19-s + (−0.667 + 0.486i)20-s + (1.16 + 0.326i)22-s + (0.278 + 0.278i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.378 - 0.925i$
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: \((2,\ 288,\ (\ :1/2),\ 0.378 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.597491 + 0.401124i\)
\(L(\frac12)\) \(\approx\) \(0.597491 + 0.401124i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.11 + 0.874i)T \)
3 \( 1 \)
good5 \( 1 + (-0.707 - 1.70i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.665 - 0.665i)T - 7iT^{2} \)
11 \( 1 + (3.69 - 1.52i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.76 - 4.26i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 3.61iT - 17T^{2} \)
19 \( 1 + (-0.194 + 0.470i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.33 - 1.33i)T + 23iT^{2} \)
29 \( 1 + (-5.73 - 2.37i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + (-0.510 - 1.23i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (1.66 + 1.66i)T + 41iT^{2} \)
43 \( 1 + (2.54 - 1.05i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 1.49iT - 47T^{2} \)
53 \( 1 + (4.59 - 1.90i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (2.04 + 4.94i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-13.7 - 5.67i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (3.40 + 1.41i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-9.66 + 9.66i)T - 71iT^{2} \)
73 \( 1 + (7.55 + 7.55i)T + 73iT^{2} \)
79 \( 1 + 17.2iT - 79T^{2} \)
83 \( 1 + (4.82 - 11.6i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-5.43 + 5.43i)T - 89iT^{2} \)
97 \( 1 - 6.15T + 97T^{2} \)
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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

−11.87854810487938376409365732652, −10.79669186188786204926893261562, −10.20529889652849604769146972395, −9.343699969812948096253347295944, −8.290438283054924265901317423170, −7.21859010507107691449178809492, −6.37985701421321306059155217729, −4.65435752643423736147289804763, −3.09605039857674866799324376282, −2.05635760932544635240374618518, 0.68225379284310050875371985005, 2.72500809066651800090528330797, 4.94410937368392307295986172728, 5.56594708052494927991541747953, 6.88290159247960029462522266464, 7.937669910486562797830729559554, 8.618967848842558355278417193053, 9.780361264620108704279472910955, 10.32801012290478483020963336254, 11.40935857645751610173088723884

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