Properties

Label 2-48-16.5-c5-0-3
Degree $2$
Conductor $48$
Sign $-0.997 + 0.0727i$
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  + (1.74 + 5.38i)2-s + (−6.36 + 6.36i)3-s + (−25.9 + 18.7i)4-s + (63.5 + 63.5i)5-s + (−45.3 − 23.1i)6-s + 39.2i·7-s + (−145. − 107. i)8-s − 81i·9-s + (−231. + 452. i)10-s + (−362. − 362. i)11-s + (45.8 − 284. i)12-s + (−607. + 607. i)13-s + (−211. + 68.2i)14-s − 809.·15-s + (321. − 972. i)16-s + 1.37e3·17-s + ⋯
L(s)  = 1  + (0.307 + 0.951i)2-s + (−0.408 + 0.408i)3-s + (−0.810 + 0.585i)4-s + (1.13 + 1.13i)5-s + (−0.514 − 0.262i)6-s + 0.302i·7-s + (−0.806 − 0.591i)8-s − 0.333i·9-s + (−0.732 + 1.43i)10-s + (−0.903 − 0.903i)11-s + (0.0919 − 0.569i)12-s + (−0.996 + 0.996i)13-s + (−0.287 + 0.0931i)14-s − 0.928·15-s + (0.314 − 0.949i)16-s + 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0727i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.997 + 0.0727i$
Analytic conductor: \(7.69842\)
Root analytic conductor: \(2.77460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :5/2),\ -0.997 + 0.0727i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0535137 - 1.46887i\)
\(L(\frac12)\) \(\approx\) \(0.0535137 - 1.46887i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.74 - 5.38i)T \)
3 \( 1 + (6.36 - 6.36i)T \)
good5 \( 1 + (-63.5 - 63.5i)T + 3.12e3iT^{2} \)
7 \( 1 - 39.2iT - 1.68e4T^{2} \)
11 \( 1 + (362. + 362. i)T + 1.61e5iT^{2} \)
13 \( 1 + (607. - 607. i)T - 3.71e5iT^{2} \)
17 \( 1 - 1.37e3T + 1.41e6T^{2} \)
19 \( 1 + (-525. + 525. i)T - 2.47e6iT^{2} \)
23 \( 1 - 2.26e3iT - 6.43e6T^{2} \)
29 \( 1 + (4.33e3 - 4.33e3i)T - 2.05e7iT^{2} \)
31 \( 1 - 7.13e3T + 2.86e7T^{2} \)
37 \( 1 + (-4.84e3 - 4.84e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 2.63e3iT - 1.15e8T^{2} \)
43 \( 1 + (-1.56e3 - 1.56e3i)T + 1.47e8iT^{2} \)
47 \( 1 + 2.12e4T + 2.29e8T^{2} \)
53 \( 1 + (-2.62e4 - 2.62e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (1.19e4 + 1.19e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (-2.11e3 + 2.11e3i)T - 8.44e8iT^{2} \)
67 \( 1 + (3.20e4 - 3.20e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 3.75e4iT - 1.80e9T^{2} \)
73 \( 1 - 692. iT - 2.07e9T^{2} \)
79 \( 1 - 3.61e4T + 3.07e9T^{2} \)
83 \( 1 + (-6.02e4 + 6.02e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 8.08e4iT - 5.58e9T^{2} \)
97 \( 1 - 9.18e4T + 8.58e9T^{2} \)
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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

−15.12258580742691331581988335323, −14.24812524754753675781858641960, −13.39379995454474778181294493002, −11.78312535639962657354250918565, −10.26903650767547646304230785922, −9.290440139142007317550573182875, −7.44128602743146550459655389351, −6.15430528537534543529606795353, −5.21343993936548684108177569497, −3.02596302043537735167040428589, 0.73716202951897314902594682681, 2.29411028566031228129323728779, 4.84375409778064122621743451067, 5.68571460285812586041292394557, 7.965131290644477709893936840608, 9.733797657424861130842355687341, 10.26929157286506516418210140627, 12.11886026207353883620352188820, 12.78742505982607963869170573748, 13.54595669321818384670583258517

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