Properties

Label 2-384-8.5-c3-0-18
Degree $2$
Conductor $384$
Sign $i$
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 + 2.82i·5-s + 14.1·7-s − 9·9-s − 20i·11-s − 39.5i·13-s + 8.48·15-s − 34·17-s + 52i·19-s − 42.4i·21-s + 62.2·23-s + 117·25-s + 27i·27-s − 200. i·29-s + 110.·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.252i·5-s + 0.763·7-s − 0.333·9-s − 0.548i·11-s − 0.844i·13-s + 0.146·15-s − 0.485·17-s + 0.627i·19-s − 0.440i·21-s + 0.564·23-s + 0.936·25-s + 0.192i·27-s − 1.28i·29-s + 0.639·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $i$
Analytic conductor: \(22.6567\)
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: \((2,\ 384,\ (\ :3/2),\ i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.786716837\)
\(L(\frac12)\) \(\approx\) \(1.786716837\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.82iT - 125T^{2} \)
7 \( 1 - 14.1T + 343T^{2} \)
11 \( 1 + 20iT - 1.33e3T^{2} \)
13 \( 1 + 39.5iT - 2.19e3T^{2} \)
17 \( 1 + 34T + 4.91e3T^{2} \)
19 \( 1 - 52iT - 6.85e3T^{2} \)
23 \( 1 - 62.2T + 1.21e4T^{2} \)
29 \( 1 + 200. iT - 2.43e4T^{2} \)
31 \( 1 - 110.T + 2.97e4T^{2} \)
37 \( 1 + 271. iT - 5.06e4T^{2} \)
41 \( 1 - 26T + 6.89e4T^{2} \)
43 \( 1 + 252iT - 7.95e4T^{2} \)
47 \( 1 + 345.T + 1.03e5T^{2} \)
53 \( 1 + 681. iT - 1.48e5T^{2} \)
59 \( 1 + 364iT - 2.05e5T^{2} \)
61 \( 1 + 735. iT - 2.26e5T^{2} \)
67 \( 1 - 628iT - 3.00e5T^{2} \)
71 \( 1 + 333.T + 3.57e5T^{2} \)
73 \( 1 + 338T + 3.89e5T^{2} \)
79 \( 1 - 789.T + 4.93e5T^{2} \)
83 \( 1 - 1.03e3iT - 5.71e5T^{2} \)
89 \( 1 + 234T + 7.04e5T^{2} \)
97 \( 1 + 178T + 9.12e5T^{2} \)
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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

−10.90779437708161555587183008371, −9.843666292882086972939124332789, −8.540323970249820788423661850559, −7.991475485402841759188535263164, −6.92835586582445081438932703400, −5.91478345948786207914828955378, −4.88424119895091956135116504667, −3.40875374196671540546513024122, −2.11915038568140354187990086097, −0.64700293773732759806390948661, 1.41718853309754787593893911810, 2.93015530507394179874910201207, 4.54507972618124683788052805180, 4.87793183203197561233041376106, 6.40113102970951909394524857088, 7.40226151026692747736564609205, 8.631381732347455679685464074418, 9.178900918635878238044875792915, 10.29018951978385116896614592549, 11.13748351854466840289004521806

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