Properties

Label 2-384-8.3-c6-0-37
Degree $2$
Conductor $384$
Sign $i$
Analytic cond. $88.3407$
Root an. cond. $9.39897$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.5·3-s − 196i·5-s + 270. i·7-s + 243·9-s + 1.72e3·11-s − 2.88e3i·13-s − 3.05e3i·15-s − 2.89e3·17-s + 9.78e3·19-s + 4.21e3i·21-s + 1.62e3i·23-s − 2.27e4·25-s + 3.78e3·27-s − 1.45e4i·29-s + 6.87e3i·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.56i·5-s + 0.787i·7-s + 0.333·9-s + 1.29·11-s − 1.31i·13-s − 0.905i·15-s − 0.589·17-s + 1.42·19-s + 0.454i·21-s + 0.133i·23-s − 1.45·25-s + 0.192·27-s − 0.598i·29-s + 0.230i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $i$
Analytic conductor: \(88.3407\)
Root analytic conductor: \(9.39897\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3),\ i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.967833748\)
\(L(\frac12)\) \(\approx\) \(2.967833748\)
\(L(4)\) 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 \( 1 - 15.5T \)
good5 \( 1 + 196iT - 1.56e4T^{2} \)
7 \( 1 - 270. iT - 1.17e5T^{2} \)
11 \( 1 - 1.72e3T + 1.77e6T^{2} \)
13 \( 1 + 2.88e3iT - 4.82e6T^{2} \)
17 \( 1 + 2.89e3T + 2.41e7T^{2} \)
19 \( 1 - 9.78e3T + 4.70e7T^{2} \)
23 \( 1 - 1.62e3iT - 1.48e8T^{2} \)
29 \( 1 + 1.45e4iT - 5.94e8T^{2} \)
31 \( 1 - 6.87e3iT - 8.87e8T^{2} \)
37 \( 1 - 1.21e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.87e4T + 4.75e9T^{2} \)
43 \( 1 - 1.35e5T + 6.32e9T^{2} \)
47 \( 1 - 1.51e5iT - 1.07e10T^{2} \)
53 \( 1 + 1.44e5iT - 2.21e10T^{2} \)
59 \( 1 + 3.33e5T + 4.21e10T^{2} \)
61 \( 1 + 3.33e5iT - 5.15e10T^{2} \)
67 \( 1 + 3.19e5T + 9.04e10T^{2} \)
71 \( 1 + 4.20e5iT - 1.28e11T^{2} \)
73 \( 1 - 6.28e5T + 1.51e11T^{2} \)
79 \( 1 - 2.91e5iT - 2.43e11T^{2} \)
83 \( 1 + 3.57e5T + 3.26e11T^{2} \)
89 \( 1 + 1.37e6T + 4.96e11T^{2} \)
97 \( 1 - 4.89e5T + 8.32e11T^{2} \)
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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

−9.588045234722053989459598115039, −9.232563188775874291836913267189, −8.398422464593708796940829041730, −7.60701801560782825167327054895, −6.09200837984116699002760829127, −5.20248663862540180314850241707, −4.21204837772251925100569426271, −2.98500428587598290081915424481, −1.57187398046978677362870380447, −0.66696729893060139318968681303, 1.24665184994561214639361025379, 2.48133115201043464525156429490, 3.58851203622710696686237322064, 4.28314039892719423353218960884, 6.12679983564161528881908034032, 7.09043334192746213042732190944, 7.32567985154722438331129218697, 8.916227655012931344384934634969, 9.606823343126129623588209043220, 10.59951580608954767289989381590

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