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Properties

Label 2-384-16.5-c3-0-12
Degree $2$
Conductor $384$
Sign $0.832 + 0.553i$
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  + (2.12 − 2.12i)3-s + (3.72 + 3.72i)5-s − 20.2i·7-s − 8.99i·9-s + (47.5 + 47.5i)11-s + (−27.8 + 27.8i)13-s + 15.8·15-s + 56.3·17-s + (66.1 − 66.1i)19-s + (−42.9 − 42.9i)21-s + 3.66i·23-s − 97.2i·25-s + (−19.0 − 19.0i)27-s + (86.8 − 86.8i)29-s − 102.·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.333 + 0.333i)5-s − 1.09i·7-s − 0.333i·9-s + (1.30 + 1.30i)11-s + (−0.594 + 0.594i)13-s + 0.271·15-s + 0.804·17-s + (0.798 − 0.798i)19-s + (−0.446 − 0.446i)21-s + 0.0332i·23-s − 0.778i·25-s + (−0.136 − 0.136i)27-s + (0.556 − 0.556i)29-s − 0.595·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.832 + 0.553i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 0.832 + 0.553i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.536192602\)
\(L(\frac12)\) \(\approx\) \(2.536192602\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.12 + 2.12i)T \)
good5 \( 1 + (-3.72 - 3.72i)T + 125iT^{2} \)
7 \( 1 + 20.2iT - 343T^{2} \)
11 \( 1 + (-47.5 - 47.5i)T + 1.33e3iT^{2} \)
13 \( 1 + (27.8 - 27.8i)T - 2.19e3iT^{2} \)
17 \( 1 - 56.3T + 4.91e3T^{2} \)
19 \( 1 + (-66.1 + 66.1i)T - 6.85e3iT^{2} \)
23 \( 1 - 3.66iT - 1.21e4T^{2} \)
29 \( 1 + (-86.8 + 86.8i)T - 2.43e4iT^{2} \)
31 \( 1 + 102.T + 2.97e4T^{2} \)
37 \( 1 + (66.8 + 66.8i)T + 5.06e4iT^{2} \)
41 \( 1 + 29.5iT - 6.89e4T^{2} \)
43 \( 1 + (-372. - 372. i)T + 7.95e4iT^{2} \)
47 \( 1 - 539.T + 1.03e5T^{2} \)
53 \( 1 + (385. + 385. i)T + 1.48e5iT^{2} \)
59 \( 1 + (71.0 + 71.0i)T + 2.05e5iT^{2} \)
61 \( 1 + (-155. + 155. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-178. + 178. i)T - 3.00e5iT^{2} \)
71 \( 1 + 483. iT - 3.57e5T^{2} \)
73 \( 1 + 908. iT - 3.89e5T^{2} \)
79 \( 1 - 1.06e3T + 4.93e5T^{2} \)
83 \( 1 + (871. - 871. i)T - 5.71e5iT^{2} \)
89 \( 1 + 185. iT - 7.04e5T^{2} \)
97 \( 1 + 725.T + 9.12e5T^{2} \)
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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

−10.75036522737427741441507639809, −9.706281724420115800017194944121, −9.291413510486750149431658454417, −7.73376259199952357803752352336, −7.11624507833378335123170404163, −6.37986771992480957482996654297, −4.72209502264879301830940454244, −3.77306609493383455493241617817, −2.29687268781580274305385166079, −1.02710248138043382676299981667, 1.23228066321105913264854089336, 2.82191495410073819974458017043, 3.80004784963686713920544880589, 5.41564458552177208150025704433, 5.83071594385045406313553259708, 7.36753968402138805169344260938, 8.548351606172322508663437304382, 9.077257020118059891615400475154, 9.880647981096434033672733621570, 10.99029147971250762013220769399

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