Properties

Label 2-384-12.11-c5-0-76
Degree $2$
Conductor $384$
Sign $-0.747 - 0.664i$
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  + (10.3 − 11.6i)3-s + 68.3i·5-s − 220. i·7-s + (−28.4 − 241. i)9-s − 127.·11-s − 822.·13-s + (796. + 707. i)15-s − 1.61e3i·17-s + 1.81e3i·19-s + (−2.57e3 − 2.28e3i)21-s + 1.27e3·23-s − 1.54e3·25-s + (−3.10e3 − 2.16e3i)27-s + 7.51e3i·29-s + 8.58e3i·31-s + ⋯
L(s)  = 1  + (0.664 − 0.747i)3-s + 1.22i·5-s − 1.70i·7-s + (−0.117 − 0.993i)9-s − 0.316·11-s − 1.34·13-s + (0.913 + 0.812i)15-s − 1.35i·17-s + 1.15i·19-s + (−1.27 − 1.13i)21-s + 0.501·23-s − 0.493·25-s + (−0.820 − 0.572i)27-s + 1.65i·29-s + 1.60i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.747 - 0.664i$
Analytic conductor: \(61.5873\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :5/2),\ -0.747 - 0.664i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.06043254474\)
\(L(\frac12)\) \(\approx\) \(0.06043254474\)
\(L(\frac{7}{2})\) 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 \( 1 + (-10.3 + 11.6i)T \)
good5 \( 1 - 68.3iT - 3.12e3T^{2} \)
7 \( 1 + 220. iT - 1.68e4T^{2} \)
11 \( 1 + 127.T + 1.61e5T^{2} \)
13 \( 1 + 822.T + 3.71e5T^{2} \)
17 \( 1 + 1.61e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.81e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.27e3T + 6.43e6T^{2} \)
29 \( 1 - 7.51e3iT - 2.05e7T^{2} \)
31 \( 1 - 8.58e3iT - 2.86e7T^{2} \)
37 \( 1 + 5.30e3T + 6.93e7T^{2} \)
41 \( 1 + 2.35e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.07e4iT - 1.47e8T^{2} \)
47 \( 1 + 4.89e3T + 2.29e8T^{2} \)
53 \( 1 - 3.95e3iT - 4.18e8T^{2} \)
59 \( 1 + 4.15e4T + 7.14e8T^{2} \)
61 \( 1 + 2.00e4T + 8.44e8T^{2} \)
67 \( 1 - 4.46e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.50e4T + 1.80e9T^{2} \)
73 \( 1 + 6.13e4T + 2.07e9T^{2} \)
79 \( 1 + 4.40e4iT - 3.07e9T^{2} \)
83 \( 1 + 1.95e3T + 3.93e9T^{2} \)
89 \( 1 + 8.97e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.32e5T + 8.58e9T^{2} \)
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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

−10.22156488980087243156301311953, −9.030797525074770996932386794861, −7.58680655429780654436676908383, −7.25163038580197674290352693028, −6.67304849407669255042049671293, −4.96275024946394186880355321121, −3.50860969091285699561270572570, −2.83111053499037717910034162101, −1.40080925535106334667625520054, −0.01269549160023239933258804334, 1.99060004275429504417903300233, 2.80646098743789170138134131567, 4.43241300028544108436997450068, 5.07464904712453197360955342148, 5.99902151808843130257449376734, 7.81834794713769036917846070957, 8.478510011488114886890481739632, 9.288090933142317598554537670225, 9.726866548491490572488902872877, 11.09188977876623644683589574687

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