Properties

Label 2-384-24.11-c5-0-48
Degree $2$
Conductor $384$
Sign $0.757 + 0.653i$
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  + (−1.14 − 15.5i)3-s + 71.6·5-s + 100. i·7-s + (−240. + 35.5i)9-s + 148. i·11-s + 381. i·13-s + (−82.0 − 1.11e3i)15-s − 2.20e3i·17-s − 1.73e3·19-s + (1.56e3 − 114. i)21-s + 3.55e3·23-s + 2.01e3·25-s + (827. + 3.69e3i)27-s + 5.77e3·29-s + 5.12e3i·31-s + ⋯
L(s)  = 1  + (−0.0733 − 0.997i)3-s + 1.28·5-s + 0.775i·7-s + (−0.989 + 0.146i)9-s + 0.369i·11-s + 0.625i·13-s + (−0.0941 − 1.27i)15-s − 1.85i·17-s − 1.10·19-s + (0.773 − 0.0568i)21-s + 1.40·23-s + 0.644·25-s + (0.218 + 0.975i)27-s + 1.27·29-s + 0.957i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.757 + 0.653i$
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: \((2,\ 384,\ (\ :5/2),\ 0.757 + 0.653i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.593966704\)
\(L(\frac12)\) \(\approx\) \(2.593966704\)
\(L(\frac{7}{2})\) 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 \( 1 + (1.14 + 15.5i)T \)
good5 \( 1 - 71.6T + 3.12e3T^{2} \)
7 \( 1 - 100. iT - 1.68e4T^{2} \)
11 \( 1 - 148. iT - 1.61e5T^{2} \)
13 \( 1 - 381. iT - 3.71e5T^{2} \)
17 \( 1 + 2.20e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.73e3T + 2.47e6T^{2} \)
23 \( 1 - 3.55e3T + 6.43e6T^{2} \)
29 \( 1 - 5.77e3T + 2.05e7T^{2} \)
31 \( 1 - 5.12e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.10e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.45e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.86e4T + 1.47e8T^{2} \)
47 \( 1 - 1.24e4T + 2.29e8T^{2} \)
53 \( 1 + 4.84e3T + 4.18e8T^{2} \)
59 \( 1 - 1.82e4iT - 7.14e8T^{2} \)
61 \( 1 - 5.45e3iT - 8.44e8T^{2} \)
67 \( 1 - 2.22e4T + 1.35e9T^{2} \)
71 \( 1 + 1.03e4T + 1.80e9T^{2} \)
73 \( 1 - 6.12e4T + 2.07e9T^{2} \)
79 \( 1 - 3.13e4iT - 3.07e9T^{2} \)
83 \( 1 + 4.66e4iT - 3.93e9T^{2} \)
89 \( 1 + 9.97e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.15e5T + 8.58e9T^{2} \)
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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

−10.44719264979742027645246844797, −9.132935903155604043934358412914, −8.914993127114715670145311540222, −7.35364653953066423146714385113, −6.62701732226068056421768162416, −5.72318065030006180520069114113, −4.85288202911751367549703886908, −2.70335816749708692938745941396, −2.13897256826841782454537786133, −0.846579627700595925245763367581, 0.910580649343563328248446807242, 2.43374198257816834878902209695, 3.69278243934284412508118473145, 4.73144913879984698324189847231, 5.84831311892865876882952371340, 6.46566842262657815490047515722, 8.102645061928035326089932506176, 8.923014561443835313519568211352, 9.928141050875134316050512450226, 10.54218993366199531405480894537

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