Properties

Label 2-384-8.5-c3-0-4
Degree $2$
Conductor $384$
Sign $-1$
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 + 17.4i·5-s + 2.99·7-s − 9·9-s + 10.6i·11-s + 43.3i·13-s − 52.2·15-s − 37.8·17-s + 79.8i·19-s + 8.97i·21-s + 191.·23-s − 178.·25-s − 27i·27-s − 138. i·29-s − 212.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.55i·5-s + 0.161·7-s − 0.333·9-s + 0.291i·11-s + 0.924i·13-s − 0.900·15-s − 0.540·17-s + 0.964i·19-s + 0.0932i·21-s + 1.73·23-s − 1.43·25-s − 0.192i·27-s − 0.889i·29-s − 1.22·31-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(2)\) \(\approx\) \(1.301716449\)
\(L(\frac12)\) \(\approx\) \(1.301716449\)
\(L(\frac{5}{2})\) 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 \( 1 - 17.4iT - 125T^{2} \)
7 \( 1 - 2.99T + 343T^{2} \)
11 \( 1 - 10.6iT - 1.33e3T^{2} \)
13 \( 1 - 43.3iT - 2.19e3T^{2} \)
17 \( 1 + 37.8T + 4.91e3T^{2} \)
19 \( 1 - 79.8iT - 6.85e3T^{2} \)
23 \( 1 - 191.T + 1.21e4T^{2} \)
29 \( 1 + 138. iT - 2.43e4T^{2} \)
31 \( 1 + 212.T + 2.97e4T^{2} \)
37 \( 1 + 270. iT - 5.06e4T^{2} \)
41 \( 1 + 441.T + 6.89e4T^{2} \)
43 \( 1 + 64.1iT - 7.95e4T^{2} \)
47 \( 1 - 436.T + 1.03e5T^{2} \)
53 \( 1 - 278. iT - 1.48e5T^{2} \)
59 \( 1 + 830. iT - 2.05e5T^{2} \)
61 \( 1 - 724. iT - 2.26e5T^{2} \)
67 \( 1 - 859. iT - 3.00e5T^{2} \)
71 \( 1 + 681.T + 3.57e5T^{2} \)
73 \( 1 + 785.T + 3.89e5T^{2} \)
79 \( 1 - 1.01e3T + 4.93e5T^{2} \)
83 \( 1 - 467. iT - 5.71e5T^{2} \)
89 \( 1 - 510.T + 7.04e5T^{2} \)
97 \( 1 + 234.T + 9.12e5T^{2} \)
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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

−11.12831043761509208697032573424, −10.59822486991303269040299840706, −9.676492913198935647638972931862, −8.768065821015767016279995017017, −7.40211723956724102847826424364, −6.75889934025324456156843934298, −5.64715816071820087814767017211, −4.28475167253485057076311849914, −3.28186833246346149849886744446, −2.07782175514304555586530936971, 0.44473068048442218055234134422, 1.50427433505096264048914611877, 3.14988076913166215555801434426, 4.80453614088276991312085473526, 5.34368668832791519203834000922, 6.69587303980055486361878103020, 7.76216707616343718977802364088, 8.750084897933369864018800060570, 9.117204001425407658768109521329, 10.57982691571019738498804222789

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