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Properties

Label 2-384-16.13-c3-0-20
Degree $2$
Conductor $384$
Sign $0.604 + 0.796i$
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 − 14.7i·7-s + 8.99i·9-s + (24.9 − 24.9i)11-s + (58.1 + 58.1i)13-s + 49.9·15-s − 75.8·17-s + (−51.8 − 51.8i)19-s + (31.2 − 31.2i)21-s − 149. i·23-s − 152. i·25-s + (−19.0 + 19.0i)27-s + (−48.5 − 48.5i)29-s + 29.6·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (1.05 − 1.05i)5-s − 0.794i·7-s + 0.333i·9-s + (0.683 − 0.683i)11-s + (1.24 + 1.24i)13-s + 0.859·15-s − 1.08·17-s + (−0.626 − 0.626i)19-s + (0.324 − 0.324i)21-s − 1.35i·23-s − 1.21i·25-s + (−0.136 + 0.136i)27-s + (−0.310 − 0.310i)29-s + 0.171·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.604 + 0.796i$
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: \((2,\ 384,\ (\ :3/2),\ 0.604 + 0.796i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.713340434\)
\(L(\frac12)\) \(\approx\) \(2.713340434\)
\(L(\frac{5}{2})\) 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 \( 1 + 14.7iT - 343T^{2} \)
11 \( 1 + (-24.9 + 24.9i)T - 1.33e3iT^{2} \)
13 \( 1 + (-58.1 - 58.1i)T + 2.19e3iT^{2} \)
17 \( 1 + 75.8T + 4.91e3T^{2} \)
19 \( 1 + (51.8 + 51.8i)T + 6.85e3iT^{2} \)
23 \( 1 + 149. iT - 1.21e4T^{2} \)
29 \( 1 + (48.5 + 48.5i)T + 2.43e4iT^{2} \)
31 \( 1 - 29.6T + 2.97e4T^{2} \)
37 \( 1 + (-147. + 147. i)T - 5.06e4iT^{2} \)
41 \( 1 - 225. iT - 6.89e4T^{2} \)
43 \( 1 + (81.7 - 81.7i)T - 7.95e4iT^{2} \)
47 \( 1 - 46.5T + 1.03e5T^{2} \)
53 \( 1 + (-156. + 156. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-238. + 238. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-594. - 594. i)T + 2.26e5iT^{2} \)
67 \( 1 + (299. + 299. i)T + 3.00e5iT^{2} \)
71 \( 1 + 693. iT - 3.57e5T^{2} \)
73 \( 1 - 462. iT - 3.89e5T^{2} \)
79 \( 1 + 878.T + 4.93e5T^{2} \)
83 \( 1 + (-926. - 926. i)T + 5.71e5iT^{2} \)
89 \( 1 + 350. iT - 7.04e5T^{2} \)
97 \( 1 + 766.T + 9.12e5T^{2} \)
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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

−10.76993390945814520942285475427, −9.670009867821419471248056505499, −8.853161497169574271982458821117, −8.523066141834851418657667753236, −6.76838571272234917943507862328, −6.04739932319376963351554148009, −4.60176952536243325832449710269, −3.98864448409023147782046211317, −2.17670818048164981271613854444, −0.931851487672843919461072415863, 1.62098024152511574265284028928, 2.58149568528294604036160220388, 3.73879681125329761750760788021, 5.59467012612068048999419244964, 6.26134148741111374604150605065, 7.13997637996929627275847482445, 8.372355280852758324112426407926, 9.205844844970857483020888205432, 10.12181681902585189937599629325, 10.94894186544243750008221750796

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