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Properties

Label 2-384-8.5-c7-0-17
Degree $2$
Conductor $384$
Sign $-i$
Analytic cond. $119.955$
Root an. cond. $10.9524$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 27i·3-s − 8.99i·5-s + 616.·7-s − 729·9-s − 3.03e3i·11-s − 1.22e4i·13-s + 242.·15-s − 3.54e4·17-s + 4.68e4i·19-s + 1.66e4i·21-s − 8.98e4·23-s + 7.80e4·25-s − 1.96e4i·27-s + 1.13e5i·29-s + 2.46e5·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.0321i·5-s + 0.679·7-s − 0.333·9-s − 0.687i·11-s − 1.54i·13-s + 0.0185·15-s − 1.75·17-s + 1.56i·19-s + 0.392i·21-s − 1.53·23-s + 0.998·25-s − 0.192i·27-s + 0.861i·29-s + 1.48·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-i$
Analytic conductor: \(119.955\)
Root analytic conductor: \(10.9524\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :7/2),\ -i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.599068911\)
\(L(\frac12)\) \(\approx\) \(1.599068911\)
\(L(\frac{9}{2})\) 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 \( 1 - 27iT \)
good5 \( 1 + 8.99iT - 7.81e4T^{2} \)
7 \( 1 - 616.T + 8.23e5T^{2} \)
11 \( 1 + 3.03e3iT - 1.94e7T^{2} \)
13 \( 1 + 1.22e4iT - 6.27e7T^{2} \)
17 \( 1 + 3.54e4T + 4.10e8T^{2} \)
19 \( 1 - 4.68e4iT - 8.93e8T^{2} \)
23 \( 1 + 8.98e4T + 3.40e9T^{2} \)
29 \( 1 - 1.13e5iT - 1.72e10T^{2} \)
31 \( 1 - 2.46e5T + 2.75e10T^{2} \)
37 \( 1 + 3.83e5iT - 9.49e10T^{2} \)
41 \( 1 - 6.38e4T + 1.94e11T^{2} \)
43 \( 1 - 6.97e5iT - 2.71e11T^{2} \)
47 \( 1 - 6.65e4T + 5.06e11T^{2} \)
53 \( 1 - 1.02e6iT - 1.17e12T^{2} \)
59 \( 1 + 1.66e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.94e6iT - 3.14e12T^{2} \)
67 \( 1 - 5.09e4iT - 6.06e12T^{2} \)
71 \( 1 - 5.04e6T + 9.09e12T^{2} \)
73 \( 1 - 5.82e6T + 1.10e13T^{2} \)
79 \( 1 - 2.30e6T + 1.92e13T^{2} \)
83 \( 1 - 3.20e6iT - 2.71e13T^{2} \)
89 \( 1 + 5.21e6T + 4.42e13T^{2} \)
97 \( 1 + 3.08e6T + 8.07e13T^{2} \)
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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

−10.58732093782049196007948462669, −9.525300950709077841860118600809, −8.337171185559852344931626276201, −8.007362953977262233436354518732, −6.41903657685689670316358446703, −5.53914972241646460744686234221, −4.54577775039963789852537023351, −3.51384040318899621799534790460, −2.33375438195196182201754403449, −0.903353429551993335403204577830, 0.38071144948694189106907705665, 1.81871405641475249808268924049, 2.43652014409741548194612165022, 4.27703744108342501089320884226, 4.85036996982731393122096968729, 6.54955717143232948529197323477, 6.84174800913581802522614845251, 8.130481617325238267990206155227, 8.884787467957550567687969388105, 9.836953334628072984921966874672

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