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Properties

Label 2-48-16.13-c5-0-7
Degree $2$
Conductor $48$
Sign $-0.840 - 0.542i$
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  + (1.92 + 5.31i)2-s + (6.36 + 6.36i)3-s + (−24.5 + 20.4i)4-s + (31.1 − 31.1i)5-s + (−21.5 + 46.1i)6-s + 246. i·7-s + (−156. − 91.2i)8-s + 81i·9-s + (225. + 105. i)10-s + (−181. + 181. i)11-s + (−286. − 25.9i)12-s + (−415. − 415. i)13-s + (−1.31e3 + 474. i)14-s + 396.·15-s + (183. − 1.00e3i)16-s + 1.42e3·17-s + ⋯
L(s)  = 1  + (0.340 + 0.940i)2-s + (0.408 + 0.408i)3-s + (−0.768 + 0.640i)4-s + (0.557 − 0.557i)5-s + (−0.244 + 0.522i)6-s + 1.90i·7-s + (−0.863 − 0.503i)8-s + 0.333i·9-s + (0.714 + 0.334i)10-s + (−0.451 + 0.451i)11-s + (−0.574 − 0.0520i)12-s + (−0.681 − 0.681i)13-s + (−1.78 + 0.647i)14-s + 0.455·15-s + (0.179 − 0.983i)16-s + 1.19·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.840 - 0.542i$
Analytic conductor: \(7.69842\)
Root analytic conductor: \(2.77460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :5/2),\ -0.840 - 0.542i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.563638 + 1.91199i\)
\(L(\frac12)\) \(\approx\) \(0.563638 + 1.91199i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.92 - 5.31i)T \)
3 \( 1 + (-6.36 - 6.36i)T \)
good5 \( 1 + (-31.1 + 31.1i)T - 3.12e3iT^{2} \)
7 \( 1 - 246. iT - 1.68e4T^{2} \)
11 \( 1 + (181. - 181. i)T - 1.61e5iT^{2} \)
13 \( 1 + (415. + 415. i)T + 3.71e5iT^{2} \)
17 \( 1 - 1.42e3T + 1.41e6T^{2} \)
19 \( 1 + (209. + 209. i)T + 2.47e6iT^{2} \)
23 \( 1 - 1.21e3iT - 6.43e6T^{2} \)
29 \( 1 + (-2.94e3 - 2.94e3i)T + 2.05e7iT^{2} \)
31 \( 1 - 7.46e3T + 2.86e7T^{2} \)
37 \( 1 + (-9.43e3 + 9.43e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 6.72e3iT - 1.15e8T^{2} \)
43 \( 1 + (-66.8 + 66.8i)T - 1.47e8iT^{2} \)
47 \( 1 - 1.85e4T + 2.29e8T^{2} \)
53 \( 1 + (1.64e4 - 1.64e4i)T - 4.18e8iT^{2} \)
59 \( 1 + (-2.68e4 + 2.68e4i)T - 7.14e8iT^{2} \)
61 \( 1 + (8.99e3 + 8.99e3i)T + 8.44e8iT^{2} \)
67 \( 1 + (4.27e4 + 4.27e4i)T + 1.35e9iT^{2} \)
71 \( 1 - 8.05e4iT - 1.80e9T^{2} \)
73 \( 1 + 2.74e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.08e4T + 3.07e9T^{2} \)
83 \( 1 + (8.88e3 + 8.88e3i)T + 3.93e9iT^{2} \)
89 \( 1 - 874. iT - 5.58e9T^{2} \)
97 \( 1 + 4.44e4T + 8.58e9T^{2} \)
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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

−15.23453254030479236977975822069, −14.29134334712772417541134406032, −12.85583564776753769317774206057, −12.17555221279735662442917927225, −9.793784485156290649380222391478, −8.942076781820335113758265038137, −7.82215877908325425980206061840, −5.77291977426519026103666743570, −5.02990607758252075708156348236, −2.76731528022207325671535825628, 0.954489459251629220779361062319, 2.81368109735810547709740948966, 4.36691866573251090152956783304, 6.44443337232562833625378560628, 7.945552570496242561024382600665, 9.891389218093973086854095118945, 10.43699581145265904828037838188, 11.85382240661224158179254123699, 13.29804262890719058199576038659, 13.93248433351370620688902057357

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