Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·3-s + 2.82·5-s − 4.89i·7-s − 2.99·9-s + 3.46i·11-s − 4.89i·15-s − 8.48·21-s + 3.00·25-s + 5.19i·27-s + 2.82·29-s − 4.89i·31-s + 5.99·33-s − 13.8i·35-s − 8.48·45-s − 16.9·49-s + ⋯
L(s)  = 1  − 0.999i·3-s + 1.26·5-s − 1.85i·7-s − 0.999·9-s + 1.04i·11-s − 1.26i·15-s − 1.85·21-s + 0.600·25-s + 0.999i·27-s + 0.525·29-s − 0.879i·31-s + 1.04·33-s − 2.34i·35-s − 1.26·45-s − 2.42·49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $i$
Motivic weight: \(1\)
Character: $\chi_{384} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10870 - 1.10870i\)
\(L(\frac12)\) \(\approx\) \(1.10870 - 1.10870i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
good5 \( 1 - 2.82T + 5T^{2} \)
7 \( 1 + 4.89iT - 7T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 + 4.89iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 14.1T + 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 - 14.6iT - 79T^{2} \)
83 \( 1 - 17.3iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−10.99610042436231843172565510573, −10.17667333838459606769819445249, −9.473631002461414628881229428724, −8.093540629927526297741136445291, −7.16049473788098287931266862280, −6.60591535003832294623708602413, −5.42512073171419559766799484655, −4.09027501861879344834821570027, −2.36969536973786420608727183049, −1.14793255303810996878577054749, 2.24922053309418090877101395844, 3.24064585936274133936720393286, 5.01786783295086349677149415785, 5.69497222128689072308810440741, 6.31888510608475523802588580068, 8.387139524508073143740744199665, 8.932902022988672480944492989716, 9.641506657524194233949913949317, 10.52407573646537156370235543594, 11.48983946172269775987873204758

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