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Properties

Label 2-48-16.5-c5-0-16
Degree $2$
Conductor $48$
Sign $-0.510 + 0.859i$
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  + (4.81 − 2.97i)2-s + (−6.36 + 6.36i)3-s + (14.3 − 28.6i)4-s + (−30.6 − 30.6i)5-s + (−11.7 + 49.5i)6-s − 92.2i·7-s + (−16.1 − 180. i)8-s − 81i·9-s + (−239. − 56.4i)10-s + (−246. − 246. i)11-s + (90.9 + 273. i)12-s + (296. − 296. i)13-s + (−274. − 444. i)14-s + 390.·15-s + (−613. − 819. i)16-s + 554.·17-s + ⋯
L(s)  = 1  + (0.850 − 0.525i)2-s + (−0.408 + 0.408i)3-s + (0.447 − 0.894i)4-s + (−0.549 − 0.549i)5-s + (−0.132 + 0.561i)6-s − 0.711i·7-s + (−0.0890 − 0.996i)8-s − 0.333i·9-s + (−0.755 − 0.178i)10-s + (−0.615 − 0.615i)11-s + (0.182 + 0.547i)12-s + (0.486 − 0.486i)13-s + (−0.374 − 0.605i)14-s + 0.448·15-s + (−0.599 − 0.800i)16-s + 0.465·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.510 + 0.859i$
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: \((2,\ 48,\ (\ :5/2),\ -0.510 + 0.859i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.910381 - 1.59898i\)
\(L(\frac12)\) \(\approx\) \(0.910381 - 1.59898i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.81 + 2.97i)T \)
3 \( 1 + (6.36 - 6.36i)T \)
good5 \( 1 + (30.6 + 30.6i)T + 3.12e3iT^{2} \)
7 \( 1 + 92.2iT - 1.68e4T^{2} \)
11 \( 1 + (246. + 246. i)T + 1.61e5iT^{2} \)
13 \( 1 + (-296. + 296. i)T - 3.71e5iT^{2} \)
17 \( 1 - 554.T + 1.41e6T^{2} \)
19 \( 1 + (574. - 574. i)T - 2.47e6iT^{2} \)
23 \( 1 - 4.12e3iT - 6.43e6T^{2} \)
29 \( 1 + (-1.67e3 + 1.67e3i)T - 2.05e7iT^{2} \)
31 \( 1 - 1.02e4T + 2.86e7T^{2} \)
37 \( 1 + (-235. - 235. i)T + 6.93e7iT^{2} \)
41 \( 1 + 542. iT - 1.15e8T^{2} \)
43 \( 1 + (-1.04e4 - 1.04e4i)T + 1.47e8iT^{2} \)
47 \( 1 + 4.00e3T + 2.29e8T^{2} \)
53 \( 1 + (1.99e4 + 1.99e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (2.99e4 + 2.99e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (-1.53e4 + 1.53e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (-5.10e3 + 5.10e3i)T - 1.35e9iT^{2} \)
71 \( 1 - 5.42e4iT - 1.80e9T^{2} \)
73 \( 1 + 8.30e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.42e4T + 3.07e9T^{2} \)
83 \( 1 + (2.87e4 - 2.87e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 4.86e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.32e5T + 8.58e9T^{2} \)
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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

−14.05357000665378140706848014164, −13.06872318440315469676700970618, −11.91130488970016499036123161727, −10.90959616373565160554476510559, −9.891981302236983632374756636202, −7.975149235450878431556552366301, −6.08086427874158911786369683383, −4.73400591384988677579702104787, −3.45215106783899589106295198414, −0.788945713611762802869316305200, 2.65624388151735211179912095070, 4.58651872781698825836834800527, 6.11426273292531588363978451585, 7.22380349315093673651238989815, 8.490802270660028260038739228218, 10.70771675529755417284446473075, 11.89270269996848716676164008040, 12.65360573584584707920629146766, 13.95221870123077107927663463506, 15.10789424740079795860702258609

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