Properties

Degree $2$
Conductor $384$
Sign $0.934 + 0.356i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.618 − 1.61i)3-s + 3.23i·5-s − 1.23i·7-s + (−2.23 − 2.00i)9-s + 5.23·11-s + 4.47·13-s + (5.23 + 2.00i)15-s − 2.47i·17-s + 0.763i·19-s + (−2.00 − 0.763i)21-s + 2.47·23-s − 5.47·25-s + (−4.61 + 2.38i)27-s + 4.76i·29-s − 5.23i·31-s + ⋯
L(s)  = 1  + (0.356 − 0.934i)3-s + 1.44i·5-s − 0.467i·7-s + (−0.745 − 0.666i)9-s + 1.57·11-s + 1.24·13-s + (1.35 + 0.516i)15-s − 0.599i·17-s + 0.175i·19-s + (−0.436 − 0.166i)21-s + 0.515·23-s − 1.09·25-s + (−0.888 + 0.458i)27-s + 0.884i·29-s − 0.940i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.934 + 0.356i$
Motivic weight: \(1\)
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.934 + 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60962 - 0.296948i\)
\(L(\frac12)\) \(\approx\) \(1.60962 - 0.296948i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.618 + 1.61i)T \)
good5 \( 1 - 3.23iT - 5T^{2} \)
7 \( 1 + 1.23iT - 7T^{2} \)
11 \( 1 - 5.23T + 11T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 + 2.47iT - 17T^{2} \)
19 \( 1 - 0.763iT - 19T^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 - 4.76iT - 29T^{2} \)
31 \( 1 + 5.23iT - 31T^{2} \)
37 \( 1 + 8.47T + 37T^{2} \)
41 \( 1 - 6.47iT - 41T^{2} \)
43 \( 1 + 7.23iT - 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 3.23iT - 53T^{2} \)
59 \( 1 + 1.23T + 59T^{2} \)
61 \( 1 + 0.472T + 61T^{2} \)
67 \( 1 - 9.70iT - 67T^{2} \)
71 \( 1 + 15.4T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 0.291iT - 79T^{2} \)
83 \( 1 - 2.76T + 83T^{2} \)
89 \( 1 - 4iT - 89T^{2} \)
97 \( 1 - 0.472T + 97T^{2} \)
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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

−11.35938087296978120152053947417, −10.56718514876331542746735153428, −9.342747798133658534827646705644, −8.460887599165852441627775716635, −7.18525770703991326931854786251, −6.79698648521432253349227155231, −5.93826331989077339506237106707, −3.86457554916008562357222693846, −3.03819750642008649545686305237, −1.45472451603873987349616811025, 1.51012220631758179412520442508, 3.52229369124652651480767571762, 4.38230861728745604736417430792, 5.38207263894208148420383547763, 6.40602062376300254341970292162, 8.160166407138318214300651917701, 8.968546524243489079159293704635, 9.110588519555404409207739862895, 10.38547358187732430650582116834, 11.45064716632547921795721179282

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