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Properties

Label 2-384-48.11-c3-0-32
Degree $2$
Conductor $384$
Sign $-0.941 + 0.336i$
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  + (−4.68 − 2.24i)3-s + (−2.69 + 2.69i)5-s + 10.6·7-s + (16.9 + 21.0i)9-s + (−29.3 − 29.3i)11-s + (7.80 − 7.80i)13-s + (18.6 − 6.58i)15-s + 13.2i·17-s + (85.6 + 85.6i)19-s + (−49.8 − 23.8i)21-s − 166. i·23-s + 110. i·25-s + (−32.3 − 136. i)27-s + (58.7 + 58.7i)29-s − 249. i·31-s + ⋯
L(s)  = 1  + (−0.902 − 0.431i)3-s + (−0.240 + 0.240i)5-s + 0.574·7-s + (0.627 + 0.778i)9-s + (−0.804 − 0.804i)11-s + (0.166 − 0.166i)13-s + (0.320 − 0.113i)15-s + 0.188i·17-s + (1.03 + 1.03i)19-s + (−0.517 − 0.247i)21-s − 1.50i·23-s + 0.884i·25-s + (−0.230 − 0.973i)27-s + (0.375 + 0.375i)29-s − 1.44i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.941 + 0.336i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ -0.941 + 0.336i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4219353092\)
\(L(\frac12)\) \(\approx\) \(0.4219353092\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.68 + 2.24i)T \)
good5 \( 1 + (2.69 - 2.69i)T - 125iT^{2} \)
7 \( 1 - 10.6T + 343T^{2} \)
11 \( 1 + (29.3 + 29.3i)T + 1.33e3iT^{2} \)
13 \( 1 + (-7.80 + 7.80i)T - 2.19e3iT^{2} \)
17 \( 1 - 13.2iT - 4.91e3T^{2} \)
19 \( 1 + (-85.6 - 85.6i)T + 6.85e3iT^{2} \)
23 \( 1 + 166. iT - 1.21e4T^{2} \)
29 \( 1 + (-58.7 - 58.7i)T + 2.43e4iT^{2} \)
31 \( 1 + 249. iT - 2.97e4T^{2} \)
37 \( 1 + (174. + 174. i)T + 5.06e4iT^{2} \)
41 \( 1 + 469.T + 6.89e4T^{2} \)
43 \( 1 + (86.4 - 86.4i)T - 7.95e4iT^{2} \)
47 \( 1 + 585.T + 1.03e5T^{2} \)
53 \( 1 + (318. - 318. i)T - 1.48e5iT^{2} \)
59 \( 1 + (273. + 273. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-270. + 270. i)T - 2.26e5iT^{2} \)
67 \( 1 + (241. + 241. i)T + 3.00e5iT^{2} \)
71 \( 1 + 203. iT - 3.57e5T^{2} \)
73 \( 1 + 47.3iT - 3.89e5T^{2} \)
79 \( 1 - 160. iT - 4.93e5T^{2} \)
83 \( 1 + (382. - 382. i)T - 5.71e5iT^{2} \)
89 \( 1 - 588.T + 7.04e5T^{2} \)
97 \( 1 + 172.T + 9.12e5T^{2} \)
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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

−10.77175961917646286943826590179, −9.867242892236207500574752818811, −8.299844685117012670159108905134, −7.76412023701898252281086199683, −6.61152618877867663425792907802, −5.64488511813766192757714831801, −4.80829378581202453908266436648, −3.29409446987341622104097728690, −1.64694772308491194446545032041, −0.16825347846210165879904953371, 1.45840626932653937204840564485, 3.33632798795930233003953072079, 4.91398744252670472740833535928, 5.04505975532246727423867083760, 6.59130204552352134508493635343, 7.47378459317695611966596209411, 8.587313494356897926469386210749, 9.756587765774837949463072314542, 10.36588500219559287166348179677, 11.56029687929369067923283808272

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