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Properties

Label 2-48-12.11-c5-0-6
Degree $2$
Conductor $48$
Sign $0.769 + 0.638i$
Analytic cond. $7.69842$
Root an. cond. $2.77460$
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  + (12 + 9.94i)3-s − 79.5i·5-s − 179. i·7-s + (45 + 238. i)9-s + 648·11-s − 242·13-s + (792 − 955. i)15-s + 318. i·17-s − 1.25e3i·19-s + (1.78e3 − 2.14e3i)21-s + 1.29e3·23-s − 3.21e3·25-s + (−1.83e3 + 3.31e3i)27-s + 1.98e3i·29-s + 3.40e3i·31-s + ⋯
L(s)  = 1  + (0.769 + 0.638i)3-s − 1.42i·5-s − 1.38i·7-s + (0.185 + 0.982i)9-s + 1.61·11-s − 0.397·13-s + (0.908 − 1.09i)15-s + 0.267i·17-s − 0.796i·19-s + (0.881 − 1.06i)21-s + 0.510·23-s − 1.02·25-s + (−0.484 + 0.874i)27-s + 0.439i·29-s + 0.635i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.769 + 0.638i$
Analytic conductor: \(7.69842\)
Root analytic conductor: \(2.77460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :5/2),\ 0.769 + 0.638i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.01212 - 0.725678i\)
\(L(\frac12)\) \(\approx\) \(2.01212 - 0.725678i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-12 - 9.94i)T \)
good5 \( 1 + 79.5iT - 3.12e3T^{2} \)
7 \( 1 + 179. iT - 1.68e4T^{2} \)
11 \( 1 - 648T + 1.61e5T^{2} \)
13 \( 1 + 242T + 3.71e5T^{2} \)
17 \( 1 - 318. iT - 1.41e6T^{2} \)
19 \( 1 + 1.25e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.29e3T + 6.43e6T^{2} \)
29 \( 1 - 1.98e3iT - 2.05e7T^{2} \)
31 \( 1 - 3.40e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.20e4T + 6.93e7T^{2} \)
41 \( 1 + 1.48e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.80e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.29e4T + 2.29e8T^{2} \)
53 \( 1 - 2.69e4iT - 4.18e8T^{2} \)
59 \( 1 - 8.42e3T + 7.14e8T^{2} \)
61 \( 1 + 2.57e4T + 8.44e8T^{2} \)
67 \( 1 - 1.02e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.57e4T + 1.80e9T^{2} \)
73 \( 1 - 2.60e4T + 2.07e9T^{2} \)
79 \( 1 - 1.91e4iT - 3.07e9T^{2} \)
83 \( 1 + 7.84e4T + 3.93e9T^{2} \)
89 \( 1 - 8.42e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.03e5T + 8.58e9T^{2} \)
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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

−14.36317740100824711612657758064, −13.57885586685863235963073695363, −12.37236898804660655551309058413, −10.78109304667281320493211053787, −9.434335189512990665071264866995, −8.669619529935670120078334942163, −7.15630205632918575835927423397, −4.82021104028938605631819022182, −3.84593347347276326113220606827, −1.19374229243361420945347647272, 2.10416020590992095632850889787, 3.42020354654293110599545564520, 6.15057481462694374652141959374, 7.13607741697556791976975282665, 8.665040761004924492090964214046, 9.755764718281155765990687166326, 11.53693801456430301554361452429, 12.32879673888747014444068254005, 13.96887580908310312042156693761, 14.71129369044257939475333131646

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