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Properties

Label 2-384-8.5-c3-0-10
Degree $2$
Conductor $384$
Sign $0.707 - 0.707i$
Analytic cond. $22.6567$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s + 18.4i·5-s + 22.4·7-s − 9·9-s − 53.6i·11-s + 7.15i·13-s + 55.2·15-s + 39.6·17-s + 125. i·19-s − 67.2i·21-s + 99.1·23-s − 214.·25-s + 27i·27-s + 205. i·29-s − 147.·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.64i·5-s + 1.21·7-s − 0.333·9-s − 1.47i·11-s + 0.152i·13-s + 0.951·15-s + 0.566·17-s + 1.51i·19-s − 0.698i·21-s + 0.898·23-s − 1.71·25-s + 0.192i·27-s + 1.31i·29-s − 0.854·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.110351356\)
\(L(\frac12)\) \(\approx\) \(2.110351356\)
\(L(\frac{5}{2})\) 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 \( 1 + 3iT \)
good5 \( 1 - 18.4iT - 125T^{2} \)
7 \( 1 - 22.4T + 343T^{2} \)
11 \( 1 + 53.6iT - 1.33e3T^{2} \)
13 \( 1 - 7.15iT - 2.19e3T^{2} \)
17 \( 1 - 39.6T + 4.91e3T^{2} \)
19 \( 1 - 125. iT - 6.85e3T^{2} \)
23 \( 1 - 99.1T + 1.21e4T^{2} \)
29 \( 1 - 205. iT - 2.43e4T^{2} \)
31 \( 1 + 147.T + 2.97e4T^{2} \)
37 \( 1 + 125. iT - 5.06e4T^{2} \)
41 \( 1 - 506.T + 6.89e4T^{2} \)
43 \( 1 - 413. iT - 7.95e4T^{2} \)
47 \( 1 - 313.T + 1.03e5T^{2} \)
53 \( 1 - 44.3iT - 1.48e5T^{2} \)
59 \( 1 - 324iT - 2.05e5T^{2} \)
61 \( 1 - 324iT - 2.26e5T^{2} \)
67 \( 1 + 464. iT - 3.00e5T^{2} \)
71 \( 1 - 1.05e3T + 3.57e5T^{2} \)
73 \( 1 + 1.02e3T + 3.89e5T^{2} \)
79 \( 1 - 602.T + 4.93e5T^{2} \)
83 \( 1 + 15.8iT - 5.71e5T^{2} \)
89 \( 1 + 381.T + 7.04e5T^{2} \)
97 \( 1 - 659.T + 9.12e5T^{2} \)
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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

−10.97072556871786719208349750635, −10.57780883473119647446396368893, −9.068982891573091221792037087242, −7.948919490273926557684207758723, −7.41307558194206554426927431279, −6.26597809018203429468873452697, −5.49833039472147976549957587842, −3.71392575480308516797721506940, −2.72957511084558171217329710992, −1.32174349401729479741596330884, 0.822364584531941637277034558000, 2.15848546524374831459178904879, 4.22144921170111837608413568024, 4.82182598867083594077417838878, 5.47240840413814229215303951401, 7.27150192568504865621404658773, 8.158428555016607450421191862890, 9.071203057850908925130135922038, 9.634963240677629992117136417457, 10.85841649326914788456082433167

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