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Properties

Label 2-384-16.11-c6-0-0
Degree $2$
Conductor $384$
Sign $-0.979 + 0.203i$
Analytic cond. $88.3407$
Root an. cond. $9.39897$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11.0 + 11.0i)3-s + (9.26 + 9.26i)5-s + 320.·7-s + 242. i·9-s + (1.49e3 − 1.49e3i)11-s + (−2.54e3 + 2.54e3i)13-s + 204. i·15-s − 7.23e3·17-s + (−4.86e3 − 4.86e3i)19-s + (3.53e3 + 3.53e3i)21-s − 1.22e4·23-s − 1.54e4i·25-s + (−2.67e3 + 2.67e3i)27-s + (−9.52e3 + 9.52e3i)29-s + 1.35e4i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.0741 + 0.0741i)5-s + 0.934·7-s + 0.333i·9-s + (1.12 − 1.12i)11-s + (−1.15 + 1.15i)13-s + 0.0605i·15-s − 1.47·17-s + (−0.709 − 0.709i)19-s + (0.381 + 0.381i)21-s − 1.00·23-s − 0.989i·25-s + (−0.136 + 0.136i)27-s + (−0.390 + 0.390i)29-s + 0.453i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.979 + 0.203i$
Analytic conductor: \(88.3407\)
Root analytic conductor: \(9.39897\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3),\ -0.979 + 0.203i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.2184699222\)
\(L(\frac12)\) \(\approx\) \(0.2184699222\)
\(L(4)\) 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 \( 1 + (-11.0 - 11.0i)T \)
good5 \( 1 + (-9.26 - 9.26i)T + 1.56e4iT^{2} \)
7 \( 1 - 320.T + 1.17e5T^{2} \)
11 \( 1 + (-1.49e3 + 1.49e3i)T - 1.77e6iT^{2} \)
13 \( 1 + (2.54e3 - 2.54e3i)T - 4.82e6iT^{2} \)
17 \( 1 + 7.23e3T + 2.41e7T^{2} \)
19 \( 1 + (4.86e3 + 4.86e3i)T + 4.70e7iT^{2} \)
23 \( 1 + 1.22e4T + 1.48e8T^{2} \)
29 \( 1 + (9.52e3 - 9.52e3i)T - 5.94e8iT^{2} \)
31 \( 1 - 1.35e4iT - 8.87e8T^{2} \)
37 \( 1 + (-964. - 964. i)T + 2.56e9iT^{2} \)
41 \( 1 - 9.46e4iT - 4.75e9T^{2} \)
43 \( 1 + (-6.14e4 + 6.14e4i)T - 6.32e9iT^{2} \)
47 \( 1 - 1.25e5iT - 1.07e10T^{2} \)
53 \( 1 + (9.27e4 + 9.27e4i)T + 2.21e10iT^{2} \)
59 \( 1 + (7.89e4 - 7.89e4i)T - 4.21e10iT^{2} \)
61 \( 1 + (1.40e4 - 1.40e4i)T - 5.15e10iT^{2} \)
67 \( 1 + (1.69e5 + 1.69e5i)T + 9.04e10iT^{2} \)
71 \( 1 + 2.39e5T + 1.28e11T^{2} \)
73 \( 1 - 6.68e5iT - 1.51e11T^{2} \)
79 \( 1 - 5.72e5iT - 2.43e11T^{2} \)
83 \( 1 + (3.60e5 + 3.60e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 1.01e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.29e6T + 8.32e11T^{2} \)
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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

−11.04810706750462456564201031172, −9.753570358189702616379179370858, −8.910870565436511551684685537832, −8.329080629040713528689922280483, −7.03476004611581553090894080434, −6.16484279941289178772874132691, −4.64778198346909477375322582387, −4.17528524732773209849529415210, −2.61611710285337810067561880184, −1.63349568912695298804153890336, 0.04049367162147880757446513334, 1.63395948277884507224717081893, 2.29613555927010345952995488842, 3.94324196634594962176840123072, 4.81693843733568527559777963333, 6.08449733292708064634383932161, 7.23295655957741272927275861654, 7.86510828621142625457818463065, 8.905362833973836883533557925714, 9.739268243560414651681228538499

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