Properties

Label 2-384-8.5-c5-0-31
Degree $2$
Conductor $384$
Sign $i$
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  − 9i·3-s + 21.1i·5-s + 241.·7-s − 81·9-s − 641. i·11-s + 316. i·13-s + 190.·15-s − 901.·17-s − 2.16e3i·19-s − 2.17e3i·21-s − 397.·23-s + 2.67e3·25-s + 729i·27-s + 5.17e3i·29-s + 2.39e3·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.377i·5-s + 1.86·7-s − 0.333·9-s − 1.59i·11-s + 0.519i·13-s + 0.218·15-s − 0.756·17-s − 1.37i·19-s − 1.07i·21-s − 0.156·23-s + 0.857·25-s + 0.192i·27-s + 1.14i·29-s + 0.446·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $i$
Analytic conductor: \(61.5873\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :5/2),\ i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.392258049\)
\(L(\frac12)\) \(\approx\) \(2.392258049\)
\(L(\frac{7}{2})\) 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 \( 1 + 9iT \)
good5 \( 1 - 21.1iT - 3.12e3T^{2} \)
7 \( 1 - 241.T + 1.68e4T^{2} \)
11 \( 1 + 641. iT - 1.61e5T^{2} \)
13 \( 1 - 316. iT - 3.71e5T^{2} \)
17 \( 1 + 901.T + 1.41e6T^{2} \)
19 \( 1 + 2.16e3iT - 2.47e6T^{2} \)
23 \( 1 + 397.T + 6.43e6T^{2} \)
29 \( 1 - 5.17e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.39e3T + 2.86e7T^{2} \)
37 \( 1 + 1.35e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.10e4T + 1.15e8T^{2} \)
43 \( 1 - 1.69e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.26e4T + 2.29e8T^{2} \)
53 \( 1 + 7.71e3iT - 4.18e8T^{2} \)
59 \( 1 + 2.86e4iT - 7.14e8T^{2} \)
61 \( 1 + 1.07e4iT - 8.44e8T^{2} \)
67 \( 1 + 3.59e4iT - 1.35e9T^{2} \)
71 \( 1 + 6.49e4T + 1.80e9T^{2} \)
73 \( 1 - 1.44e4T + 2.07e9T^{2} \)
79 \( 1 + 3.79e4T + 3.07e9T^{2} \)
83 \( 1 + 5.07e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.70e4T + 5.58e9T^{2} \)
97 \( 1 + 9.59e4T + 8.58e9T^{2} \)
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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

−10.90758076914406928350189801504, −8.941697667429781039750273224370, −8.523035768219967460398686886225, −7.48536962769780489119335716529, −6.59697542534670208454272088565, −5.42467678267272954468688015392, −4.47408677364286972414757351082, −2.94029617897843226032839702022, −1.77339007071295132442182071968, −0.62309427377989873352467210785, 1.29630723101957361365019757221, 2.31512082627375235329583057311, 4.18676849993654726697773853355, 4.72192665039343375291689531522, 5.63752594572619486195491275260, 7.18880400886525177031291320522, 8.113877595315779888750208371220, 8.768528204633877625467470116621, 10.04796219880112800636581879969, 10.56529930536390442853266416126

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