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Properties

Label 2-384-12.11-c3-0-3
Degree $2$
Conductor $384$
Sign $-0.874 + 0.485i$
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  + (2.52 + 4.54i)3-s + 8.01i·5-s + 12.6i·7-s + (−14.2 + 22.9i)9-s − 37.7·11-s − 60.0·13-s + (−36.3 + 20.2i)15-s − 37.0i·17-s − 127. i·19-s + (−57.2 + 31.7i)21-s + 56.9·23-s + 60.8·25-s + (−140. − 7.09i)27-s + 220. i·29-s − 2.26i·31-s + ⋯
L(s)  = 1  + (0.485 + 0.874i)3-s + 0.716i·5-s + 0.680i·7-s + (−0.528 + 0.848i)9-s − 1.03·11-s − 1.28·13-s + (−0.626 + 0.347i)15-s − 0.528i·17-s − 1.54i·19-s + (−0.594 + 0.330i)21-s + 0.516·23-s + 0.486·25-s + (−0.998 − 0.0505i)27-s + 1.41i·29-s − 0.0130i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 + 0.485i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.874 + 0.485i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ -0.874 + 0.485i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8124691901\)
\(L(\frac12)\) \(\approx\) \(0.8124691901\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.52 - 4.54i)T \)
good5 \( 1 - 8.01iT - 125T^{2} \)
7 \( 1 - 12.6iT - 343T^{2} \)
11 \( 1 + 37.7T + 1.33e3T^{2} \)
13 \( 1 + 60.0T + 2.19e3T^{2} \)
17 \( 1 + 37.0iT - 4.91e3T^{2} \)
19 \( 1 + 127. iT - 6.85e3T^{2} \)
23 \( 1 - 56.9T + 1.21e4T^{2} \)
29 \( 1 - 220. iT - 2.43e4T^{2} \)
31 \( 1 + 2.26iT - 2.97e4T^{2} \)
37 \( 1 - 166.T + 5.06e4T^{2} \)
41 \( 1 + 154. iT - 6.89e4T^{2} \)
43 \( 1 + 53.4iT - 7.95e4T^{2} \)
47 \( 1 + 591.T + 1.03e5T^{2} \)
53 \( 1 - 538. iT - 1.48e5T^{2} \)
59 \( 1 + 586.T + 2.05e5T^{2} \)
61 \( 1 + 431.T + 2.26e5T^{2} \)
67 \( 1 - 175. iT - 3.00e5T^{2} \)
71 \( 1 + 29.5T + 3.57e5T^{2} \)
73 \( 1 - 937.T + 3.89e5T^{2} \)
79 \( 1 + 409. iT - 4.93e5T^{2} \)
83 \( 1 + 1.29e3T + 5.71e5T^{2} \)
89 \( 1 - 921. iT - 7.04e5T^{2} \)
97 \( 1 + 1.64e3T + 9.12e5T^{2} \)
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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

−11.10482148646362577857775897795, −10.59622371691850067259079984070, −9.545157554532382854187547773116, −8.923393237951200899097215609183, −7.74273188577935488941487854258, −6.87830739097687560317643964393, −5.30892916712912952015767047515, −4.73812001151229301002697143283, −2.96512216282600226694190871208, −2.57227595126236422050990667753, 0.24240105266584600544725486656, 1.63410523156739380169538700587, 2.94144671792100317205647136762, 4.36521471365607059327195392673, 5.54082882260927159484628055140, 6.71782307552371624582900705510, 7.87248017500174336039999995850, 8.111584692239191233737242729022, 9.485769030888751325429552652085, 10.21735297158632310614777743821

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