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Properties

Label 2-384-16.13-c3-0-6
Degree $2$
Conductor $384$
Sign $-0.995 + 0.0937i$
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 + (−11.7 + 11.7i)5-s + 12.5i·7-s + 8.99i·9-s + (17.0 − 17.0i)11-s + (49.2 + 49.2i)13-s − 50.0·15-s − 51.8·17-s + (11.6 + 11.6i)19-s + (−26.6 + 26.6i)21-s + 74.5i·23-s − 153. i·25-s + (−19.0 + 19.0i)27-s + (−211. − 211. i)29-s − 326.·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−1.05 + 1.05i)5-s + 0.679i·7-s + 0.333i·9-s + (0.466 − 0.466i)11-s + (1.05 + 1.05i)13-s − 0.861·15-s − 0.739·17-s + (0.140 + 0.140i)19-s + (−0.277 + 0.277i)21-s + 0.675i·23-s − 1.22i·25-s + (−0.136 + 0.136i)27-s + (−1.35 − 1.35i)29-s − 1.88·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0937i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.995 + 0.0937i$
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: \((2,\ 384,\ (\ :3/2),\ -0.995 + 0.0937i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9963546651\)
\(L(\frac12)\) \(\approx\) \(0.9963546651\)
\(L(\frac{5}{2})\) 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 \( 1 + (11.7 - 11.7i)T - 125iT^{2} \)
7 \( 1 - 12.5iT - 343T^{2} \)
11 \( 1 + (-17.0 + 17.0i)T - 1.33e3iT^{2} \)
13 \( 1 + (-49.2 - 49.2i)T + 2.19e3iT^{2} \)
17 \( 1 + 51.8T + 4.91e3T^{2} \)
19 \( 1 + (-11.6 - 11.6i)T + 6.85e3iT^{2} \)
23 \( 1 - 74.5iT - 1.21e4T^{2} \)
29 \( 1 + (211. + 211. i)T + 2.43e4iT^{2} \)
31 \( 1 + 326.T + 2.97e4T^{2} \)
37 \( 1 + (110. - 110. i)T - 5.06e4iT^{2} \)
41 \( 1 + 348. iT - 6.89e4T^{2} \)
43 \( 1 + (205. - 205. i)T - 7.95e4iT^{2} \)
47 \( 1 - 254.T + 1.03e5T^{2} \)
53 \( 1 + (-225. + 225. i)T - 1.48e5iT^{2} \)
59 \( 1 + (285. - 285. i)T - 2.05e5iT^{2} \)
61 \( 1 + (286. + 286. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-627. - 627. i)T + 3.00e5iT^{2} \)
71 \( 1 + 274. iT - 3.57e5T^{2} \)
73 \( 1 + 298. iT - 3.89e5T^{2} \)
79 \( 1 + 175.T + 4.93e5T^{2} \)
83 \( 1 + (125. + 125. i)T + 5.71e5iT^{2} \)
89 \( 1 - 900. iT - 7.04e5T^{2} \)
97 \( 1 - 5.27T + 9.12e5T^{2} \)
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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

−11.34434954203646467780447702250, −10.73232256739577971905678281299, −9.339614000145776906198305253429, −8.748567453018220451323521026227, −7.66896778070805335342484976445, −6.76008314386691448456518824896, −5.68634793901213485369011354201, −4.00592545999057963212980468804, −3.52601873835357555615158143298, −2.07475934296082975613946380732, 0.32678179185614225834234978290, 1.51503856799786654746520734667, 3.47466306970095274449801390665, 4.23040490648204796860098870266, 5.46437898739181367316256673419, 6.93896799936856582228862842623, 7.67151398115947048231299159486, 8.603476701199291342883507708237, 9.165173823935523160781776066752, 10.63137581755140309657229959730

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