Properties

Degree $2$
Conductor $384$
Sign $-0.850 + 0.526i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.57 + 2.55i)3-s + 1.31i·5-s − 10.2·7-s + (−4.01 + 8.05i)9-s − 16.6i·11-s − 18.7·13-s + (−3.35 + 2.07i)15-s + 4.38i·17-s − 11.5·19-s + (−16.1 − 26.1i)21-s − 16.7i·23-s + 23.2·25-s + (−26.8 + 2.46i)27-s + 12.5i·29-s − 20.3·31-s + ⋯
L(s)  = 1  + (0.526 + 0.850i)3-s + 0.263i·5-s − 1.46·7-s + (−0.446 + 0.894i)9-s − 1.51i·11-s − 1.44·13-s + (−0.223 + 0.138i)15-s + 0.257i·17-s − 0.608·19-s + (−0.769 − 1.24i)21-s − 0.728i·23-s + 0.930·25-s + (−0.995 + 0.0912i)27-s + 0.432i·29-s − 0.655·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.850 + 0.526i$
Motivic weight: \(2\)
Character: $\chi_{384} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ -0.850 + 0.526i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0406659 - 0.143017i\)
\(L(\frac12)\) \(\approx\) \(0.0406659 - 0.143017i\)
\(L(2)\) 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 \( 1 + (-1.57 - 2.55i)T \)
good5 \( 1 - 1.31iT - 25T^{2} \)
7 \( 1 + 10.2T + 49T^{2} \)
11 \( 1 + 16.6iT - 121T^{2} \)
13 \( 1 + 18.7T + 169T^{2} \)
17 \( 1 - 4.38iT - 289T^{2} \)
19 \( 1 + 11.5T + 361T^{2} \)
23 \( 1 + 16.7iT - 529T^{2} \)
29 \( 1 - 12.5iT - 841T^{2} \)
31 \( 1 + 20.3T + 961T^{2} \)
37 \( 1 - 18.5T + 1.36e3T^{2} \)
41 \( 1 - 78.6iT - 1.68e3T^{2} \)
43 \( 1 + 36.4T + 1.84e3T^{2} \)
47 \( 1 - 19.9iT - 2.20e3T^{2} \)
53 \( 1 + 81.3iT - 2.80e3T^{2} \)
59 \( 1 + 29.9iT - 3.48e3T^{2} \)
61 \( 1 + 72.0T + 3.72e3T^{2} \)
67 \( 1 + 56.3T + 4.48e3T^{2} \)
71 \( 1 - 136. iT - 5.04e3T^{2} \)
73 \( 1 + 80.8T + 5.32e3T^{2} \)
79 \( 1 + 86.0T + 6.24e3T^{2} \)
83 \( 1 + 80.4iT - 6.88e3T^{2} \)
89 \( 1 - 131. iT - 7.92e3T^{2} \)
97 \( 1 + 20.4T + 9.40e3T^{2} \)
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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

−11.44401145442328722800078755262, −10.52382365846519314723175371446, −9.861704386868474311025695199271, −9.031606886796445262993851941159, −8.182793936142801254636720523620, −6.87238080804761372095129296075, −5.92562530107165593876239471010, −4.67193701519684812621453599601, −3.35773487564911303084005536261, −2.73872639691586088489323272443, 0.05599237165318034808520399415, 2.03229589058712001218997123777, 3.10226060078879742967827719988, 4.52756943728239607390323743159, 5.94764181229705895160719171722, 7.14900669431720426726594002075, 7.35343919529039675124070871419, 8.917437740719497629773502108930, 9.536870607827285848755591773563, 10.30577795051897527947655453507

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