Properties

Label 2-384-24.11-c5-0-47
Degree $2$
Conductor $384$
Sign $0.941 + 0.336i$
Analytic cond. $61.5873$
Root an. cond. $7.84776$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (14.0 − 6.67i)3-s − 28.8·5-s + 44.6i·7-s + (153. − 188. i)9-s − 152. i·11-s + 578. i·13-s + (−407. + 192. i)15-s + 82.5i·17-s + 1.23e3·19-s + (297. + 628. i)21-s + 2.59e3·23-s − 2.28e3·25-s + (914. − 3.67e3i)27-s − 53.2·29-s + 4.03e3i·31-s + ⋯
L(s)  = 1  + (0.903 − 0.428i)3-s − 0.516·5-s + 0.344i·7-s + (0.633 − 0.773i)9-s − 0.379i·11-s + 0.948i·13-s + (−0.467 + 0.221i)15-s + 0.0692i·17-s + 0.783·19-s + (0.147 + 0.311i)21-s + 1.02·23-s − 0.732·25-s + (0.241 − 0.970i)27-s − 0.0117·29-s + 0.753i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.941 + 0.336i$
Analytic conductor: \(61.5873\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :5/2),\ 0.941 + 0.336i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.820624622\)
\(L(\frac12)\) \(\approx\) \(2.820624622\)
\(L(\frac{7}{2})\) 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 \( 1 + (-14.0 + 6.67i)T \)
good5 \( 1 + 28.8T + 3.12e3T^{2} \)
7 \( 1 - 44.6iT - 1.68e4T^{2} \)
11 \( 1 + 152. iT - 1.61e5T^{2} \)
13 \( 1 - 578. iT - 3.71e5T^{2} \)
17 \( 1 - 82.5iT - 1.41e6T^{2} \)
19 \( 1 - 1.23e3T + 2.47e6T^{2} \)
23 \( 1 - 2.59e3T + 6.43e6T^{2} \)
29 \( 1 + 53.2T + 2.05e7T^{2} \)
31 \( 1 - 4.03e3iT - 2.86e7T^{2} \)
37 \( 1 + 3.92e3iT - 6.93e7T^{2} \)
41 \( 1 + 4.85e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.45e4T + 1.47e8T^{2} \)
47 \( 1 - 1.09e4T + 2.29e8T^{2} \)
53 \( 1 - 3.57e4T + 4.18e8T^{2} \)
59 \( 1 + 2.01e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.82e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.40e3T + 1.35e9T^{2} \)
71 \( 1 + 4.44e4T + 1.80e9T^{2} \)
73 \( 1 - 1.75e4T + 2.07e9T^{2} \)
79 \( 1 + 7.34e4iT - 3.07e9T^{2} \)
83 \( 1 - 3.70e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.06e5iT - 5.58e9T^{2} \)
97 \( 1 - 3.28e4T + 8.58e9T^{2} \)
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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

−10.41338627914239642880511879903, −9.176022465538597936674726155025, −8.778585210595626421414320910158, −7.61532150122911111194602755466, −6.98815002713722347594835798852, −5.73100723563212983230003151801, −4.28966884489925749741392911744, −3.31919037434093679064098450072, −2.18227632639131502351980367372, −0.866899051404468740509449749019, 0.892167344251910323733284869328, 2.49142352314212915642421790168, 3.52389741444785399193213622352, 4.44388260836839144133496387933, 5.57663180763959707263838847365, 7.22131444820114833697170562361, 7.73769465880066666722757216032, 8.722390007754111547581134343620, 9.660881018440933791216090341075, 10.41895264662640758817254725809

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