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Properties

Label 2-384-12.11-c3-0-46
Degree $2$
Conductor $384$
Sign $-0.585 - 0.810i$
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  + (−3.04 − 4.21i)3-s − 9.33i·5-s − 36.3i·7-s + (−8.50 + 25.6i)9-s − 48.4·11-s − 25.8·13-s + (−39.3 + 28.3i)15-s − 74.2i·17-s − 82.9i·19-s + (−153. + 110. i)21-s + 179.·23-s + 37.8·25-s + (133. − 42.1i)27-s + 122. i·29-s + 64.1i·31-s + ⋯
L(s)  = 1  + (−0.585 − 0.810i)3-s − 0.834i·5-s − 1.96i·7-s + (−0.314 + 0.949i)9-s − 1.32·11-s − 0.552·13-s + (−0.676 + 0.488i)15-s − 1.05i·17-s − 1.00i·19-s + (−1.59 + 1.14i)21-s + 1.62·23-s + 0.303·25-s + (0.953 − 0.300i)27-s + 0.783i·29-s + 0.371i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.585 - 0.810i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ -0.585 - 0.810i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7565916768\)
\(L(\frac12)\) \(\approx\) \(0.7565916768\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.04 + 4.21i)T \)
good5 \( 1 + 9.33iT - 125T^{2} \)
7 \( 1 + 36.3iT - 343T^{2} \)
11 \( 1 + 48.4T + 1.33e3T^{2} \)
13 \( 1 + 25.8T + 2.19e3T^{2} \)
17 \( 1 + 74.2iT - 4.91e3T^{2} \)
19 \( 1 + 82.9iT - 6.85e3T^{2} \)
23 \( 1 - 179.T + 1.21e4T^{2} \)
29 \( 1 - 122. iT - 2.43e4T^{2} \)
31 \( 1 - 64.1iT - 2.97e4T^{2} \)
37 \( 1 + 5.01T + 5.06e4T^{2} \)
41 \( 1 - 325. iT - 6.89e4T^{2} \)
43 \( 1 + 321. iT - 7.95e4T^{2} \)
47 \( 1 + 95.9T + 1.03e5T^{2} \)
53 \( 1 - 185. iT - 1.48e5T^{2} \)
59 \( 1 + 226.T + 2.05e5T^{2} \)
61 \( 1 + 198.T + 2.26e5T^{2} \)
67 \( 1 - 23.9iT - 3.00e5T^{2} \)
71 \( 1 - 399.T + 3.57e5T^{2} \)
73 \( 1 - 669.T + 3.89e5T^{2} \)
79 \( 1 - 229. iT - 4.93e5T^{2} \)
83 \( 1 - 321.T + 5.71e5T^{2} \)
89 \( 1 - 131. iT - 7.04e5T^{2} \)
97 \( 1 - 136.T + 9.12e5T^{2} \)
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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

−10.67485195903582969366180250331, −9.464661330094905694945897415698, −8.195505857884239568584624955526, −7.26591031364701475934152352290, −6.88962023318025201003136750891, −5.05096003993291965521181278251, −4.80962236881927138394355542411, −2.90328920234238547439817110311, −1.10229598842869200659052347122, −0.31689440930255174536722834201, 2.36591239401093730944587077174, 3.27430065085454333894978675883, 4.93042285564182989071723139188, 5.65965701257344831381533042608, 6.45057684370911310604332945593, 7.920200929774603136963842631561, 8.890753183105400935114665792679, 9.811974799608779480885610880763, 10.64121820504824578736856892931, 11.34589823057966290798189678630

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