Properties

Label 2-384-8.3-c6-0-22
Degree $2$
Conductor $384$
Sign $1$
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 + 85.3i·5-s + 511. i·7-s + 243·9-s − 1.21e3·11-s − 2.74e3i·13-s − 1.33e3i·15-s − 8.52e3·17-s + 1.04e4·19-s − 7.97e3i·21-s − 4.30e3i·23-s + 8.33e3·25-s − 3.78e3·27-s − 4.24e4i·29-s − 5.78e4i·31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.683i·5-s + 1.49i·7-s + 0.333·9-s − 0.911·11-s − 1.24i·13-s − 0.394i·15-s − 1.73·17-s + 1.52·19-s − 0.861i·21-s − 0.353i·23-s + 0.533·25-s − 0.192·27-s − 1.74i·29-s − 1.94i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $1$
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: \((2,\ 384,\ (\ :3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.197860972\)
\(L(\frac12)\) \(\approx\) \(1.197860972\)
\(L(4)\) 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 \( 1 - 85.3iT - 1.56e4T^{2} \)
7 \( 1 - 511. iT - 1.17e5T^{2} \)
11 \( 1 + 1.21e3T + 1.77e6T^{2} \)
13 \( 1 + 2.74e3iT - 4.82e6T^{2} \)
17 \( 1 + 8.52e3T + 2.41e7T^{2} \)
19 \( 1 - 1.04e4T + 4.70e7T^{2} \)
23 \( 1 + 4.30e3iT - 1.48e8T^{2} \)
29 \( 1 + 4.24e4iT - 5.94e8T^{2} \)
31 \( 1 + 5.78e4iT - 8.87e8T^{2} \)
37 \( 1 - 8.35e4iT - 2.56e9T^{2} \)
41 \( 1 + 7.01e4T + 4.75e9T^{2} \)
43 \( 1 - 2.87e4T + 6.32e9T^{2} \)
47 \( 1 - 5.87e4iT - 1.07e10T^{2} \)
53 \( 1 - 1.11e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.18e5T + 4.21e10T^{2} \)
61 \( 1 - 5.31e4iT - 5.15e10T^{2} \)
67 \( 1 + 4.21e4T + 9.04e10T^{2} \)
71 \( 1 - 3.51e5iT - 1.28e11T^{2} \)
73 \( 1 + 9.12e4T + 1.51e11T^{2} \)
79 \( 1 + 7.75e5iT - 2.43e11T^{2} \)
83 \( 1 - 6.69e5T + 3.26e11T^{2} \)
89 \( 1 + 5.04e5T + 4.96e11T^{2} \)
97 \( 1 - 1.03e6T + 8.32e11T^{2} \)
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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

−10.38583735821116325625102096292, −9.564007459796920943882468441858, −8.420320122935036009627373222795, −7.54658155783951975024059437825, −6.30029337528688721623078335370, −5.64536761638620984889328231142, −4.69068407054840652988985417961, −2.96632240982487670020575703611, −2.31224649468550840717115733339, −0.44615790281898027238989922607, 0.68351135096152561464569701172, 1.71262745565371291798896467002, 3.51444542148108119948737122130, 4.64418222973527092986126829118, 5.22048977560910381797077246815, 6.89311983174028681564246373059, 7.15351754881531266290951427925, 8.579912672705271429785277022496, 9.430423054311882863736960228779, 10.58915822570211048233408848748

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