Properties

Label 2-384-12.11-c5-0-12
Degree $2$
Conductor $384$
Sign $-0.734 + 0.679i$
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  + (10.5 + 11.4i)3-s + 101. i·5-s + 13.9i·7-s + (−18.9 + 242. i)9-s − 445.·11-s + 217.·13-s + (−1.16e3 + 1.07e3i)15-s − 1.43e3i·17-s + 2.61e3i·19-s + (−159. + 147. i)21-s − 696.·23-s − 7.26e3·25-s + (−2.97e3 + 2.34e3i)27-s + 4.60e3i·29-s − 8.33e3i·31-s + ⋯
L(s)  = 1  + (0.679 + 0.734i)3-s + 1.82i·5-s + 0.107i·7-s + (−0.0778 + 0.996i)9-s − 1.11·11-s + 0.356·13-s + (−1.33 + 1.23i)15-s − 1.20i·17-s + 1.65i·19-s + (−0.0789 + 0.0730i)21-s − 0.274·23-s − 2.32·25-s + (−0.784 + 0.619i)27-s + 1.01i·29-s − 1.55i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 + 0.679i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.734 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.734 + 0.679i$
Analytic conductor: \(61.5873\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :5/2),\ -0.734 + 0.679i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.444133755\)
\(L(\frac12)\) \(\approx\) \(1.444133755\)
\(L(\frac{7}{2})\) 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 \( 1 + (-10.5 - 11.4i)T \)
good5 \( 1 - 101. iT - 3.12e3T^{2} \)
7 \( 1 - 13.9iT - 1.68e4T^{2} \)
11 \( 1 + 445.T + 1.61e5T^{2} \)
13 \( 1 - 217.T + 3.71e5T^{2} \)
17 \( 1 + 1.43e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.61e3iT - 2.47e6T^{2} \)
23 \( 1 + 696.T + 6.43e6T^{2} \)
29 \( 1 - 4.60e3iT - 2.05e7T^{2} \)
31 \( 1 + 8.33e3iT - 2.86e7T^{2} \)
37 \( 1 - 6.92e3T + 6.93e7T^{2} \)
41 \( 1 - 1.16e4iT - 1.15e8T^{2} \)
43 \( 1 - 6.05e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.30e4T + 2.29e8T^{2} \)
53 \( 1 + 3.78e4iT - 4.18e8T^{2} \)
59 \( 1 + 4.11e3T + 7.14e8T^{2} \)
61 \( 1 - 203.T + 8.44e8T^{2} \)
67 \( 1 - 1.28e3iT - 1.35e9T^{2} \)
71 \( 1 - 4.54e4T + 1.80e9T^{2} \)
73 \( 1 - 4.24e4T + 2.07e9T^{2} \)
79 \( 1 + 2.85e4iT - 3.07e9T^{2} \)
83 \( 1 + 7.61e3T + 3.93e9T^{2} \)
89 \( 1 - 1.52e4iT - 5.58e9T^{2} \)
97 \( 1 + 8.78e4T + 8.58e9T^{2} \)
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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

−10.91988456163244923218097413352, −10.11160716138129136532970425949, −9.608726433626470894744137756858, −8.081232278825760383293915376562, −7.59212666326358514398848216890, −6.37935071130650014859454397854, −5.28890664038313554018140750785, −3.84098906185728560203819870330, −2.99570677233485009012964224041, −2.20744328732978592326559843117, 0.31931483484182881792786362875, 1.27938982775671147536617523203, 2.44713519363347119374417755344, 3.96406227070619688367702004979, 5.01701803132040979999446430132, 6.04771474902009150525538937411, 7.37963813306724487427459548763, 8.319637658433491971457575643707, 8.734742220441454165333230285596, 9.654841369186260131887122380376

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