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Properties

Label 2-384-48.11-c3-0-19
Degree $2$
Conductor $384$
Sign $0.386 - 0.922i$
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  + (5.19 + 0.0749i)3-s + (−5.37 + 5.37i)5-s + 14.8·7-s + (26.9 + 0.779i)9-s + (30.0 + 30.0i)11-s + (−61.5 + 61.5i)13-s + (−28.3 + 27.5i)15-s − 48.8i·17-s + (−7.45 − 7.45i)19-s + (77.1 + 1.11i)21-s − 43.0i·23-s + 67.1i·25-s + (140. + 6.07i)27-s + (−32.9 − 32.9i)29-s + 173. i·31-s + ⋯
L(s)  = 1  + (0.999 + 0.0144i)3-s + (−0.480 + 0.480i)5-s + 0.802·7-s + (0.999 + 0.0288i)9-s + (0.823 + 0.823i)11-s + (−1.31 + 1.31i)13-s + (−0.487 + 0.473i)15-s − 0.696i·17-s + (−0.0900 − 0.0900i)19-s + (0.802 + 0.0115i)21-s − 0.390i·23-s + 0.537i·25-s + (0.999 + 0.0432i)27-s + (−0.211 − 0.211i)29-s + 1.00i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.386 - 0.922i$
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: \((2,\ 384,\ (\ :3/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.626335496\)
\(L(\frac12)\) \(\approx\) \(2.626335496\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.19 - 0.0749i)T \)
good5 \( 1 + (5.37 - 5.37i)T - 125iT^{2} \)
7 \( 1 - 14.8T + 343T^{2} \)
11 \( 1 + (-30.0 - 30.0i)T + 1.33e3iT^{2} \)
13 \( 1 + (61.5 - 61.5i)T - 2.19e3iT^{2} \)
17 \( 1 + 48.8iT - 4.91e3T^{2} \)
19 \( 1 + (7.45 + 7.45i)T + 6.85e3iT^{2} \)
23 \( 1 + 43.0iT - 1.21e4T^{2} \)
29 \( 1 + (32.9 + 32.9i)T + 2.43e4iT^{2} \)
31 \( 1 - 173. iT - 2.97e4T^{2} \)
37 \( 1 + (-177. - 177. i)T + 5.06e4iT^{2} \)
41 \( 1 - 454.T + 6.89e4T^{2} \)
43 \( 1 + (239. - 239. i)T - 7.95e4iT^{2} \)
47 \( 1 + 30.4T + 1.03e5T^{2} \)
53 \( 1 + (-235. + 235. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-260. - 260. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-388. + 388. i)T - 2.26e5iT^{2} \)
67 \( 1 + (334. + 334. i)T + 3.00e5iT^{2} \)
71 \( 1 - 522. iT - 3.57e5T^{2} \)
73 \( 1 - 689. iT - 3.89e5T^{2} \)
79 \( 1 + 692. iT - 4.93e5T^{2} \)
83 \( 1 + (677. - 677. i)T - 5.71e5iT^{2} \)
89 \( 1 + 261.T + 7.04e5T^{2} \)
97 \( 1 + 641.T + 9.12e5T^{2} \)
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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

−11.24567819030861862475000481872, −9.866945517367348897150669170597, −9.340222766303756169944100717289, −8.273735828386852961302515505279, −7.26333881640113254370128015209, −6.84980336512992641044443167197, −4.79963809430415978107318791381, −4.15529365245612011865649123817, −2.72507009181024644670959759964, −1.63583272729272236150211076191, 0.832307684406772952613382257239, 2.33147778013738174227815344717, 3.65894589809337613297518761951, 4.59141385708286193417608087258, 5.83728919281794729688806928864, 7.39510638058206738995037386482, 8.019097914529249542276098825060, 8.721657290278943112506735196811, 9.690617191994238856041530238017, 10.67808539034426324777410839850

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