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Properties

Label 2-384-24.11-c5-0-9
Degree $2$
Conductor $384$
Sign $-0.0258 - 0.999i$
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  + (10.7 − 11.3i)3-s + 4.25·5-s − 187. i·7-s + (−12.5 − 242. i)9-s + 712. i·11-s + 804. i·13-s + (45.6 − 48.0i)15-s + 309. i·17-s − 2.46e3·19-s + (−2.11e3 − 2.01e3i)21-s − 3.06e3·23-s − 3.10e3·25-s + (−2.87e3 − 2.46e3i)27-s + 6.32e3·29-s + 9.00e3i·31-s + ⋯
L(s)  = 1  + (0.688 − 0.725i)3-s + 0.0760·5-s − 1.44i·7-s + (−0.0516 − 0.998i)9-s + 1.77i·11-s + 1.31i·13-s + (0.0523 − 0.0551i)15-s + 0.259i·17-s − 1.56·19-s + (−1.04 − 0.995i)21-s − 1.20·23-s − 0.994·25-s + (−0.759 − 0.650i)27-s + 1.39·29-s + 1.68i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0258 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0258 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.0258 - 0.999i$
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: \((2,\ 384,\ (\ :5/2),\ -0.0258 - 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9829283971\)
\(L(\frac12)\) \(\approx\) \(0.9829283971\)
\(L(\frac{7}{2})\) 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 \( 1 + (-10.7 + 11.3i)T \)
good5 \( 1 - 4.25T + 3.12e3T^{2} \)
7 \( 1 + 187. iT - 1.68e4T^{2} \)
11 \( 1 - 712. iT - 1.61e5T^{2} \)
13 \( 1 - 804. iT - 3.71e5T^{2} \)
17 \( 1 - 309. iT - 1.41e6T^{2} \)
19 \( 1 + 2.46e3T + 2.47e6T^{2} \)
23 \( 1 + 3.06e3T + 6.43e6T^{2} \)
29 \( 1 - 6.32e3T + 2.05e7T^{2} \)
31 \( 1 - 9.00e3iT - 2.86e7T^{2} \)
37 \( 1 - 7.35e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.03e4iT - 1.15e8T^{2} \)
43 \( 1 + 3.83e3T + 1.47e8T^{2} \)
47 \( 1 - 1.11e4T + 2.29e8T^{2} \)
53 \( 1 - 7.39e3T + 4.18e8T^{2} \)
59 \( 1 + 9.68e3iT - 7.14e8T^{2} \)
61 \( 1 - 1.54e4iT - 8.44e8T^{2} \)
67 \( 1 + 3.40e4T + 1.35e9T^{2} \)
71 \( 1 + 1.74e4T + 1.80e9T^{2} \)
73 \( 1 - 2.32e4T + 2.07e9T^{2} \)
79 \( 1 + 4.10e4iT - 3.07e9T^{2} \)
83 \( 1 + 2.07e3iT - 3.93e9T^{2} \)
89 \( 1 - 1.36e5iT - 5.58e9T^{2} \)
97 \( 1 + 9.40e4T + 8.58e9T^{2} \)
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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

−10.47965397464341799786360643539, −9.926723820497383244802469705934, −8.821553129594334403738932876071, −7.84349901552650249626046618157, −6.94620510831386555435505620813, −6.51860037341798631638378724337, −4.47863247292827545287804516796, −3.93304208974148259824064010435, −2.20236428673013320577504313445, −1.43876899244852278002573325529, 0.20365487869245021306567121808, 2.26525561153157688252981144007, 3.01912904812145426058623407428, 4.19087436479398840102566802472, 5.63964878654921093408922742045, 6.01290822147833714544308776016, 8.110934270861411636613882795707, 8.325887078765461117840804451130, 9.268259186269626516338104655886, 10.23413566787487016326059814086

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