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Properties

Label 2-288-32.29-c1-0-2
Degree 22
Conductor 288288
Sign 0.378−0.925i0.378 - 0.925i
Analytic cond. 2.299692.29969
Root an. cond. 1.516471.51647
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(288s/2 ΓC(s) L(s)=((0.378−0.925i) Λ‾(2−s)\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}
Λ(s)=(288s/2 ΓC(s+1/2) L(s)=((0.378−0.925i) Λ‾(1−s)\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: 22
Conductor: 288288    =    25⋅322^{5} \cdot 3^{2}
Sign: 0.378−0.925i0.378 - 0.925i
Analytic conductor: 2.299692.29969
Root analytic conductor: 1.516471.51647
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ288(253,⋅)\chi_{288} (253, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 288, ( :1/2), 0.378−0.925i)(2,\ 288,\ (\ :1/2),\ 0.378 - 0.925i)

Particular Values

L(1)L(1) ≈\approx 0.597491+0.401124i0.597491 + 0.401124i
L(12)L(\frac12) ≈\approx 0.597491+0.401124i0.597491 + 0.401124i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(1.11+0.874i)T 1 + (1.11 + 0.874i)T
3 1 1
good5 1+(−0.707−1.70i)T+(−3.53+3.53i)T2 1 + (-0.707 - 1.70i)T + (-3.53 + 3.53i)T^{2}
7 1+(0.665−0.665i)T−7iT2 1 + (0.665 - 0.665i)T - 7iT^{2}
11 1+(3.69−1.52i)T+(7.77−7.77i)T2 1 + (3.69 - 1.52i)T + (7.77 - 7.77i)T^{2}
13 1+(1.76−4.26i)T+(−9.19−9.19i)T2 1 + (1.76 - 4.26i)T + (-9.19 - 9.19i)T^{2}
17 1−3.61iT−17T2 1 - 3.61iT - 17T^{2}
19 1+(−0.194+0.470i)T+(−13.4−13.4i)T2 1 + (-0.194 + 0.470i)T + (-13.4 - 13.4i)T^{2}
23 1+(−1.33−1.33i)T+23iT2 1 + (-1.33 - 1.33i)T + 23iT^{2}
29 1+(−5.73−2.37i)T+(20.5+20.5i)T2 1 + (-5.73 - 2.37i)T + (20.5 + 20.5i)T^{2}
31 1−1.17T+31T2 1 - 1.17T + 31T^{2}
37 1+(−0.510−1.23i)T+(−26.1+26.1i)T2 1 + (-0.510 - 1.23i)T + (-26.1 + 26.1i)T^{2}
41 1+(1.66+1.66i)T+41iT2 1 + (1.66 + 1.66i)T + 41iT^{2}
43 1+(2.54−1.05i)T+(30.4−30.4i)T2 1 + (2.54 - 1.05i)T + (30.4 - 30.4i)T^{2}
47 1+1.49iT−47T2 1 + 1.49iT - 47T^{2}
53 1+(4.59−1.90i)T+(37.4−37.4i)T2 1 + (4.59 - 1.90i)T + (37.4 - 37.4i)T^{2}
59 1+(2.04+4.94i)T+(−41.7+41.7i)T2 1 + (2.04 + 4.94i)T + (-41.7 + 41.7i)T^{2}
61 1+(−13.7−5.67i)T+(43.1+43.1i)T2 1 + (-13.7 - 5.67i)T + (43.1 + 43.1i)T^{2}
67 1+(3.40+1.41i)T+(47.3+47.3i)T2 1 + (3.40 + 1.41i)T + (47.3 + 47.3i)T^{2}
71 1+(−9.66+9.66i)T−71iT2 1 + (-9.66 + 9.66i)T - 71iT^{2}
73 1+(7.55+7.55i)T+73iT2 1 + (7.55 + 7.55i)T + 73iT^{2}
79 1+17.2iT−79T2 1 + 17.2iT - 79T^{2}
83 1+(4.82−11.6i)T+(−58.6−58.6i)T2 1 + (4.82 - 11.6i)T + (-58.6 - 58.6i)T^{2}
89 1+(−5.43+5.43i)T−89iT2 1 + (-5.43 + 5.43i)T - 89iT^{2}
97 1−6.15T+97T2 1 - 6.15T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line