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Properties

Label 2-384-8.5-c9-0-23
Degree $2$
Conductor $384$
Sign $0.707 - 0.707i$
Analytic cond. $197.773$
Root an. cond. $14.0632$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 81i·3-s − 332. i·5-s − 3.96e3·7-s − 6.56e3·9-s − 3.23e3i·11-s + 1.15e5i·13-s + 2.69e4·15-s − 5.62e5·17-s − 1.04e6i·19-s − 3.21e5i·21-s − 1.24e6·23-s + 1.84e6·25-s − 5.31e5i·27-s + 1.09e6i·29-s + 8.23e6·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.238i·5-s − 0.624·7-s − 0.333·9-s − 0.0666i·11-s + 1.12i·13-s + 0.137·15-s − 1.63·17-s − 1.84i·19-s − 0.360i·21-s − 0.931·23-s + 0.943·25-s − 0.192i·27-s + 0.287i·29-s + 1.60·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(197.773\)
Root analytic conductor: \(14.0632\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :9/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.239275349\)
\(L(\frac12)\) \(\approx\) \(1.239275349\)
\(L(\frac{11}{2})\) 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 \( 1 - 81iT \)
good5 \( 1 + 332. iT - 1.95e6T^{2} \)
7 \( 1 + 3.96e3T + 4.03e7T^{2} \)
11 \( 1 + 3.23e3iT - 2.35e9T^{2} \)
13 \( 1 - 1.15e5iT - 1.06e10T^{2} \)
17 \( 1 + 5.62e5T + 1.18e11T^{2} \)
19 \( 1 + 1.04e6iT - 3.22e11T^{2} \)
23 \( 1 + 1.24e6T + 1.80e12T^{2} \)
29 \( 1 - 1.09e6iT - 1.45e13T^{2} \)
31 \( 1 - 8.23e6T + 2.64e13T^{2} \)
37 \( 1 + 1.93e7iT - 1.29e14T^{2} \)
41 \( 1 + 1.92e7T + 3.27e14T^{2} \)
43 \( 1 - 3.19e7iT - 5.02e14T^{2} \)
47 \( 1 + 4.55e7T + 1.11e15T^{2} \)
53 \( 1 - 5.98e7iT - 3.29e15T^{2} \)
59 \( 1 + 2.37e7iT - 8.66e15T^{2} \)
61 \( 1 + 1.57e8iT - 1.16e16T^{2} \)
67 \( 1 - 1.10e8iT - 2.72e16T^{2} \)
71 \( 1 - 2.79e8T + 4.58e16T^{2} \)
73 \( 1 - 5.92e7T + 5.88e16T^{2} \)
79 \( 1 + 1.15e8T + 1.19e17T^{2} \)
83 \( 1 + 4.47e5iT - 1.86e17T^{2} \)
89 \( 1 - 7.69e8T + 3.50e17T^{2} \)
97 \( 1 + 2.94e8T + 7.60e17T^{2} \)
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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

−9.722129022462039657661418275697, −9.141695749384517779162804035701, −8.376022664960557245847188583310, −6.85791158785434298827291802022, −6.37626710119482342805299191937, −4.85633745100420659950222593770, −4.34041490095512152014286083097, −3.05569869715466078959751666775, −2.06386846417384631700448397468, −0.51734743020907516934071964455, 0.39785434427655394389408292742, 1.65258088374124635642849237308, 2.73446800213243560972148015914, 3.68198076169117847869990973446, 5.00704850019468352790421780884, 6.23016366037341001866520584965, 6.69301017890894715568710593080, 7.988235757859654605739957199018, 8.516282440429725678314544705341, 9.931240352584177041937676037084

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