Properties

Label 2-384-8.5-c9-0-16
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 − 1.42e3i·5-s + 6.38e3·7-s − 6.56e3·9-s + 4.07e3i·11-s − 1.54e5i·13-s − 1.15e5·15-s + 2.55e5·17-s + 1.01e6i·19-s − 5.16e5i·21-s − 1.56e6·23-s − 8.63e4·25-s + 5.31e5i·27-s + 7.26e6i·29-s − 6.11e5·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.02i·5-s + 1.00·7-s − 0.333·9-s + 0.0838i·11-s − 1.49i·13-s − 0.589·15-s + 0.742·17-s + 1.79i·19-s − 0.580i·21-s − 1.16·23-s − 0.0442·25-s + 0.192i·27-s + 1.90i·29-s − 0.119·31-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

\(L(5)\) \(\approx\) \(1.553481709\)
\(L(\frac12)\) \(\approx\) \(1.553481709\)
\(L(\frac{11}{2})\) 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 \( 1 + 1.42e3iT - 1.95e6T^{2} \)
7 \( 1 - 6.38e3T + 4.03e7T^{2} \)
11 \( 1 - 4.07e3iT - 2.35e9T^{2} \)
13 \( 1 + 1.54e5iT - 1.06e10T^{2} \)
17 \( 1 - 2.55e5T + 1.18e11T^{2} \)
19 \( 1 - 1.01e6iT - 3.22e11T^{2} \)
23 \( 1 + 1.56e6T + 1.80e12T^{2} \)
29 \( 1 - 7.26e6iT - 1.45e13T^{2} \)
31 \( 1 + 6.11e5T + 2.64e13T^{2} \)
37 \( 1 + 4.64e5iT - 1.29e14T^{2} \)
41 \( 1 - 2.04e7T + 3.27e14T^{2} \)
43 \( 1 - 2.95e7iT - 5.02e14T^{2} \)
47 \( 1 - 1.31e6T + 1.11e15T^{2} \)
53 \( 1 - 5.76e7iT - 3.29e15T^{2} \)
59 \( 1 - 6.35e7iT - 8.66e15T^{2} \)
61 \( 1 + 5.01e7iT - 1.16e16T^{2} \)
67 \( 1 - 2.90e8iT - 2.72e16T^{2} \)
71 \( 1 + 3.18e8T + 4.58e16T^{2} \)
73 \( 1 + 3.49e8T + 5.88e16T^{2} \)
79 \( 1 + 4.29e8T + 1.19e17T^{2} \)
83 \( 1 - 3.40e8iT - 1.86e17T^{2} \)
89 \( 1 - 5.24e8T + 3.50e17T^{2} \)
97 \( 1 + 9.61e8T + 7.60e17T^{2} \)
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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

−9.988940545557056081805541790070, −8.716431774847270578924255260333, −8.044047936888361224117968905427, −7.52025829276237178316769768231, −5.86533929774821676769163087088, −5.36863062837891679248077033515, −4.24307591277655742724695494639, −2.96798691907025031669734554047, −1.47443600822487002897145608814, −1.13410334799128374293977990323, 0.28033205758775022062158513425, 1.84755788523286563154487448434, 2.72314116443819701668315608989, 3.99576460012520932920175991025, 4.72768992009079690473607078486, 5.95013227780702922922768655085, 6.92310779384410871818945153124, 7.82118547702766743579109774622, 8.890478187381416279297326731886, 9.755580221687614544707739322857

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