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Properties

Label 2-384-16.13-c3-0-9
Degree $2$
Conductor $384$
Sign $0.696 - 0.717i$
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  + (−2.12 − 2.12i)3-s + (−0.706 + 0.706i)5-s + 4.44i·7-s + 8.99i·9-s + (17.7 − 17.7i)11-s + (17.7 + 17.7i)13-s + 2.99·15-s − 105.·17-s + (−40.2 − 40.2i)19-s + (9.42 − 9.42i)21-s + 42.9i·23-s + 124. i·25-s + (19.0 − 19.0i)27-s + (185. + 185. i)29-s + 291.·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.0631 + 0.0631i)5-s + 0.239i·7-s + 0.333i·9-s + (0.485 − 0.485i)11-s + (0.377 + 0.377i)13-s + 0.0516·15-s − 1.50·17-s + (−0.486 − 0.486i)19-s + (0.0978 − 0.0978i)21-s + 0.389i·23-s + 0.992i·25-s + (0.136 − 0.136i)27-s + (1.18 + 1.18i)29-s + 1.68·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.696 - 0.717i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 0.696 - 0.717i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.351942966\)
\(L(\frac12)\) \(\approx\) \(1.351942966\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.12 + 2.12i)T \)
good5 \( 1 + (0.706 - 0.706i)T - 125iT^{2} \)
7 \( 1 - 4.44iT - 343T^{2} \)
11 \( 1 + (-17.7 + 17.7i)T - 1.33e3iT^{2} \)
13 \( 1 + (-17.7 - 17.7i)T + 2.19e3iT^{2} \)
17 \( 1 + 105.T + 4.91e3T^{2} \)
19 \( 1 + (40.2 + 40.2i)T + 6.85e3iT^{2} \)
23 \( 1 - 42.9iT - 1.21e4T^{2} \)
29 \( 1 + (-185. - 185. i)T + 2.43e4iT^{2} \)
31 \( 1 - 291.T + 2.97e4T^{2} \)
37 \( 1 + (151. - 151. i)T - 5.06e4iT^{2} \)
41 \( 1 + 60.3iT - 6.89e4T^{2} \)
43 \( 1 + (120. - 120. i)T - 7.95e4iT^{2} \)
47 \( 1 - 500.T + 1.03e5T^{2} \)
53 \( 1 + (192. - 192. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-500. + 500. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-166. - 166. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-575. - 575. i)T + 3.00e5iT^{2} \)
71 \( 1 - 457. iT - 3.57e5T^{2} \)
73 \( 1 - 1.08e3iT - 3.89e5T^{2} \)
79 \( 1 + 544.T + 4.93e5T^{2} \)
83 \( 1 + (48.6 + 48.6i)T + 5.71e5iT^{2} \)
89 \( 1 + 43.6iT - 7.04e5T^{2} \)
97 \( 1 - 690.T + 9.12e5T^{2} \)
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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

−11.23040030818508390154887588662, −10.28269619456180592504330430295, −8.968018046406244119256443723761, −8.438021136232237137507525084899, −6.95992538028416343788925081002, −6.46610073180951203564437787334, −5.24414931315363161535495914882, −4.10311749672839333156181751114, −2.60818976807497724483600534059, −1.13391508816204614964635711620, 0.56896733034769752911647824158, 2.35684463954149462662430305478, 4.01920128657025663166255076672, 4.67436195273425814353719700768, 6.11659456556549885351829514358, 6.77401507540302414905066815002, 8.155397314603452779981096673058, 8.947767776207527817246175202202, 10.12771270903553454012410365122, 10.63176123108243624000717213183

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