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Properties

Label 2-384-48.35-c3-0-21
Degree $2$
Conductor $384$
Sign $0.944 - 0.328i$
Analytic cond. $22.6567$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.24 + 4.68i)3-s + (2.69 + 2.69i)5-s + 10.6·7-s + (−16.9 − 21.0i)9-s + (29.3 − 29.3i)11-s + (7.80 + 7.80i)13-s + (−18.6 + 6.58i)15-s + 13.2i·17-s + (85.6 − 85.6i)19-s + (−23.8 + 49.8i)21-s − 166. i·23-s − 110. i·25-s + (136. − 32.3i)27-s + (−58.7 + 58.7i)29-s + 249. i·31-s + ⋯
L(s)  = 1  + (−0.431 + 0.902i)3-s + (0.240 + 0.240i)5-s + 0.574·7-s + (−0.627 − 0.778i)9-s + (0.804 − 0.804i)11-s + (0.166 + 0.166i)13-s + (−0.320 + 0.113i)15-s + 0.188i·17-s + (1.03 − 1.03i)19-s + (−0.247 + 0.517i)21-s − 1.50i·23-s − 0.884i·25-s + (0.973 − 0.230i)27-s + (−0.375 + 0.375i)29-s + 1.44i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.944 - 0.328i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 0.944 - 0.328i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.880807830\)
\(L(\frac12)\) \(\approx\) \(1.880807830\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.24 - 4.68i)T \)
good5 \( 1 + (-2.69 - 2.69i)T + 125iT^{2} \)
7 \( 1 - 10.6T + 343T^{2} \)
11 \( 1 + (-29.3 + 29.3i)T - 1.33e3iT^{2} \)
13 \( 1 + (-7.80 - 7.80i)T + 2.19e3iT^{2} \)
17 \( 1 - 13.2iT - 4.91e3T^{2} \)
19 \( 1 + (-85.6 + 85.6i)T - 6.85e3iT^{2} \)
23 \( 1 + 166. iT - 1.21e4T^{2} \)
29 \( 1 + (58.7 - 58.7i)T - 2.43e4iT^{2} \)
31 \( 1 - 249. iT - 2.97e4T^{2} \)
37 \( 1 + (174. - 174. i)T - 5.06e4iT^{2} \)
41 \( 1 - 469.T + 6.89e4T^{2} \)
43 \( 1 + (86.4 + 86.4i)T + 7.95e4iT^{2} \)
47 \( 1 - 585.T + 1.03e5T^{2} \)
53 \( 1 + (-318. - 318. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-273. + 273. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-270. - 270. i)T + 2.26e5iT^{2} \)
67 \( 1 + (241. - 241. i)T - 3.00e5iT^{2} \)
71 \( 1 + 203. iT - 3.57e5T^{2} \)
73 \( 1 - 47.3iT - 3.89e5T^{2} \)
79 \( 1 + 160. iT - 4.93e5T^{2} \)
83 \( 1 + (-382. - 382. i)T + 5.71e5iT^{2} \)
89 \( 1 + 588.T + 7.04e5T^{2} \)
97 \( 1 + 172.T + 9.12e5T^{2} \)
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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

−10.91046353534486231931550103599, −10.23896469157713382502714725385, −9.058576417549139776942816688282, −8.555525334366149655036731662458, −6.98448832081861814407491171810, −6.07036182233775757668896097744, −5.04390692225795855657072637731, −4.05863784985328047759274217347, −2.82634182488081625924530542461, −0.874580196888121864900093756389, 1.10204938732545036055330629712, 2.04904708114488535180648887453, 3.84702802206659616574283728596, 5.28945063233080015538161341982, 5.94355867992442345405375876293, 7.32607320211845037077959672595, 7.71792789695169605888334496531, 9.057654391382654935351058443474, 9.881874558984051132337192374640, 11.21735409773899180579086060758

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