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Properties

Label 2-384-24.11-c5-0-73
Degree $2$
Conductor $384$
Sign $-0.947 - 0.320i$
Analytic cond. $61.5873$
Root an. cond. $7.84776$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.91 − 13.9i)3-s + 56.7·5-s − 130. i·7-s + (−147. + 193. i)9-s − 484. i·11-s − 555. i·13-s + (−392. − 792. i)15-s + 1.11e3i·17-s − 741.·19-s + (−1.82e3 + 900. i)21-s + 459.·23-s + 94.6·25-s + (3.71e3 + 725. i)27-s − 3.06e3·29-s − 5.98e3i·31-s + ⋯
L(s)  = 1  + (−0.443 − 0.896i)3-s + 1.01·5-s − 1.00i·7-s + (−0.606 + 0.794i)9-s − 1.20i·11-s − 0.912i·13-s + (−0.450 − 0.909i)15-s + 0.931i·17-s − 0.470·19-s + (−0.900 + 0.445i)21-s + 0.181·23-s + 0.0302·25-s + (0.981 + 0.191i)27-s − 0.675·29-s − 1.11i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.947 - 0.320i$
Analytic conductor: \(61.5873\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :5/2),\ -0.947 - 0.320i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.199065238\)
\(L(\frac12)\) \(\approx\) \(1.199065238\)
\(L(\frac{7}{2})\) 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 \( 1 + (6.91 + 13.9i)T \)
good5 \( 1 - 56.7T + 3.12e3T^{2} \)
7 \( 1 + 130. iT - 1.68e4T^{2} \)
11 \( 1 + 484. iT - 1.61e5T^{2} \)
13 \( 1 + 555. iT - 3.71e5T^{2} \)
17 \( 1 - 1.11e3iT - 1.41e6T^{2} \)
19 \( 1 + 741.T + 2.47e6T^{2} \)
23 \( 1 - 459.T + 6.43e6T^{2} \)
29 \( 1 + 3.06e3T + 2.05e7T^{2} \)
31 \( 1 + 5.98e3iT - 2.86e7T^{2} \)
37 \( 1 + 3.46e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.49e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.68e4T + 1.47e8T^{2} \)
47 \( 1 + 1.96e4T + 2.29e8T^{2} \)
53 \( 1 - 3.22e4T + 4.18e8T^{2} \)
59 \( 1 - 5.25e3iT - 7.14e8T^{2} \)
61 \( 1 - 5.36e4iT - 8.44e8T^{2} \)
67 \( 1 + 6.66e4T + 1.35e9T^{2} \)
71 \( 1 - 3.31e4T + 1.80e9T^{2} \)
73 \( 1 + 8.03e4T + 2.07e9T^{2} \)
79 \( 1 + 1.94e4iT - 3.07e9T^{2} \)
83 \( 1 - 2.28e4iT - 3.93e9T^{2} \)
89 \( 1 + 3.35e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.46e5T + 8.58e9T^{2} \)
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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

−10.43037159302300283921285608261, −8.995608082686420148624094056628, −8.009042012459933799285007491171, −7.14741802101704325788197918798, −5.94157979055639300986137206033, −5.66368906538107833483130682372, −3.94945327393155907643762295853, −2.50759780788835467292687446002, −1.28528886559054021303299340311, −0.31083006779718080514185659693, 1.72799789958512163129926410197, 2.80151874475490053321799989901, 4.39676387224630862004677838747, 5.19482768785960865077007832602, 6.08550049281165850993456598244, 7.01429204607477890131226605539, 8.654246808974564376061424693799, 9.502947924775318844163747486315, 9.806577653392712371495165688889, 10.97878268870553600121590125439

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