Properties

Label 2-384-8.3-c8-0-3
Degree $2$
Conductor $384$
Sign $-1$
Analytic cond. $156.433$
Root an. cond. $12.5073$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 46.7·3-s − 402. i·5-s + 1.93e3i·7-s + 2.18e3·9-s − 2.01e4·11-s + 4.37e4i·13-s + 1.88e4i·15-s + 1.01e5·17-s − 1.39e4·19-s − 9.05e4i·21-s + 4.33e5i·23-s + 2.28e5·25-s − 1.02e5·27-s + 4.67e5i·29-s − 1.48e6i·31-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.643i·5-s + 0.806i·7-s + 0.333·9-s − 1.37·11-s + 1.53i·13-s + 0.371i·15-s + 1.21·17-s − 0.106·19-s − 0.465i·21-s + 1.54i·23-s + 0.585·25-s − 0.192·27-s + 0.661i·29-s − 1.60i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-1$
Analytic conductor: \(156.433\)
Root analytic conductor: \(12.5073\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :4),\ -1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.5149433971\)
\(L(\frac12)\) \(\approx\) \(0.5149433971\)
\(L(5)\) 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 \( 1 + 46.7T \)
good5 \( 1 + 402. iT - 3.90e5T^{2} \)
7 \( 1 - 1.93e3iT - 5.76e6T^{2} \)
11 \( 1 + 2.01e4T + 2.14e8T^{2} \)
13 \( 1 - 4.37e4iT - 8.15e8T^{2} \)
17 \( 1 - 1.01e5T + 6.97e9T^{2} \)
19 \( 1 + 1.39e4T + 1.69e10T^{2} \)
23 \( 1 - 4.33e5iT - 7.83e10T^{2} \)
29 \( 1 - 4.67e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.48e6iT - 8.52e11T^{2} \)
37 \( 1 - 2.57e6iT - 3.51e12T^{2} \)
41 \( 1 + 3.84e6T + 7.98e12T^{2} \)
43 \( 1 + 1.88e6T + 1.16e13T^{2} \)
47 \( 1 + 5.93e6iT - 2.38e13T^{2} \)
53 \( 1 + 8.07e6iT - 6.22e13T^{2} \)
59 \( 1 - 1.78e7T + 1.46e14T^{2} \)
61 \( 1 - 2.11e7iT - 1.91e14T^{2} \)
67 \( 1 + 7.11e6T + 4.06e14T^{2} \)
71 \( 1 + 1.36e7iT - 6.45e14T^{2} \)
73 \( 1 - 3.97e7T + 8.06e14T^{2} \)
79 \( 1 + 8.29e5iT - 1.51e15T^{2} \)
83 \( 1 - 1.45e7T + 2.25e15T^{2} \)
89 \( 1 - 7.77e7T + 3.93e15T^{2} \)
97 \( 1 + 9.95e7T + 7.83e15T^{2} \)
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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

−10.35176919828107477546802678314, −9.568679491265864093492301852558, −8.635307211795300336908829519020, −7.70570578615857027004473811592, −6.61228212567918492556042791717, −5.39605415491624907449580649003, −5.06291054141117203518176672380, −3.65490845555828967169420686169, −2.26886703675750946904848903658, −1.22586384975650234467532100407, 0.13201552623571281877056210231, 0.937539374920302640220203046122, 2.60275500150708612911232422905, 3.45649288420660646513651124220, 4.86487020364051001749545760665, 5.62424063328870319559041792716, 6.74927458155260127168213137441, 7.61193244652065412064571631847, 8.335383262685301298300720386246, 10.03040219611834651885041176408

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