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Properties

Label 2-96-24.11-c7-0-25
Degree $2$
Conductor $96$
Sign $-0.801 - 0.598i$
Analytic cond. $29.9889$
Root an. cond. $5.47621$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9 − 45.8i)3-s − 97.9·5-s − 1.54e3i·7-s + (−2.02e3 − 826. i)9-s − 1.78e3i·11-s + 1.01e4i·13-s + (−881. + 4.49e3i)15-s − 2.75e4i·17-s − 1.15e4·19-s + (−7.10e4 − 1.39e4i)21-s + 5.55e4·23-s − 6.85e4·25-s + (−5.61e4 + 8.54e4i)27-s + 5.47e4·29-s + 7.16e4i·31-s + ⋯
L(s)  = 1  + (0.192 − 0.981i)3-s − 0.350·5-s − 1.70i·7-s + (−0.925 − 0.377i)9-s − 0.404i·11-s + 1.28i·13-s + (−0.0674 + 0.343i)15-s − 1.36i·17-s − 0.386·19-s + (−1.67 − 0.328i)21-s + 0.951·23-s − 0.877·25-s + (−0.548 + 0.835i)27-s + 0.417·29-s + 0.432i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $-0.801 - 0.598i$
Analytic conductor: \(29.9889\)
Root analytic conductor: \(5.47621\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 96,\ (\ :7/2),\ -0.801 - 0.598i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.252247 + 0.759286i\)
\(L(\frac12)\) \(\approx\) \(0.252247 + 0.759286i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-9 + 45.8i)T \)
good5 \( 1 + 97.9T + 7.81e4T^{2} \)
7 \( 1 + 1.54e3iT - 8.23e5T^{2} \)
11 \( 1 + 1.78e3iT - 1.94e7T^{2} \)
13 \( 1 - 1.01e4iT - 6.27e7T^{2} \)
17 \( 1 + 2.75e4iT - 4.10e8T^{2} \)
19 \( 1 + 1.15e4T + 8.93e8T^{2} \)
23 \( 1 - 5.55e4T + 3.40e9T^{2} \)
29 \( 1 - 5.47e4T + 1.72e10T^{2} \)
31 \( 1 - 7.16e4iT - 2.75e10T^{2} \)
37 \( 1 - 2.82e5iT - 9.49e10T^{2} \)
41 \( 1 - 4.03e5iT - 1.94e11T^{2} \)
43 \( 1 + 4.95e5T + 2.71e11T^{2} \)
47 \( 1 + 1.13e6T + 5.06e11T^{2} \)
53 \( 1 - 5.72e5T + 1.17e12T^{2} \)
59 \( 1 + 1.44e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.78e6iT - 3.14e12T^{2} \)
67 \( 1 + 1.40e6T + 6.06e12T^{2} \)
71 \( 1 - 3.56e6T + 9.09e12T^{2} \)
73 \( 1 + 2.22e6T + 1.10e13T^{2} \)
79 \( 1 + 5.59e6iT - 1.92e13T^{2} \)
83 \( 1 - 3.01e6iT - 2.71e13T^{2} \)
89 \( 1 + 5.84e6iT - 4.42e13T^{2} \)
97 \( 1 - 6.86e6T + 8.07e13T^{2} \)
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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

−11.81809389842121003001574992382, −11.13988970267976402693628937662, −9.688895487203944314748452882608, −8.337172658564417872274879303416, −7.22930475705132557905327978913, −6.61633567047793226136889426074, −4.62412682558281947705684209117, −3.22718521945595389105221681005, −1.41791723942763422291340263866, −0.24506555241710578022399184665, 2.30376178945926633369953622321, 3.56505535258397094000711449875, 5.08538470560778546916329153949, 6.00482654465997007927742548066, 8.061600118158867151724760129549, 8.803662035707899252174752756311, 9.930764373153661229265300651291, 10.99188347048721533944994740461, 12.10957984834322197755300639793, 13.02183763344601987676688129011

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