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Properties

Label 2-48-16.5-c5-0-19
Degree $2$
Conductor $48$
Sign $-0.999 - 0.0299i$
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  + (2.70 − 4.96i)2-s + (6.36 − 6.36i)3-s + (−17.3 − 26.8i)4-s + (−55.5 − 55.5i)5-s + (−14.4 − 48.8i)6-s + 156. i·7-s + (−180. + 13.6i)8-s − 81i·9-s + (−426. + 125. i)10-s + (−88.6 − 88.6i)11-s + (−281. − 60.4i)12-s + (714. − 714. i)13-s + (776. + 422. i)14-s − 707.·15-s + (−420. + 933. i)16-s − 1.56e3·17-s + ⋯
L(s)  = 1  + (0.477 − 0.878i)2-s + (0.408 − 0.408i)3-s + (−0.543 − 0.839i)4-s + (−0.994 − 0.994i)5-s + (−0.163 − 0.553i)6-s + 1.20i·7-s + (−0.997 + 0.0756i)8-s − 0.333i·9-s + (−1.34 + 0.398i)10-s + (−0.220 − 0.220i)11-s + (−0.564 − 0.121i)12-s + (1.17 − 1.17i)13-s + (1.05 + 0.575i)14-s − 0.812·15-s + (−0.410 + 0.912i)16-s − 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0299i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.999 - 0.0299i$
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: \((2,\ 48,\ (\ :5/2),\ -0.999 - 0.0299i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0224323 + 1.49892i\)
\(L(\frac12)\) \(\approx\) \(0.0224323 + 1.49892i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.70 + 4.96i)T \)
3 \( 1 + (-6.36 + 6.36i)T \)
good5 \( 1 + (55.5 + 55.5i)T + 3.12e3iT^{2} \)
7 \( 1 - 156. iT - 1.68e4T^{2} \)
11 \( 1 + (88.6 + 88.6i)T + 1.61e5iT^{2} \)
13 \( 1 + (-714. + 714. i)T - 3.71e5iT^{2} \)
17 \( 1 + 1.56e3T + 1.41e6T^{2} \)
19 \( 1 + (-347. + 347. i)T - 2.47e6iT^{2} \)
23 \( 1 + 3.35e3iT - 6.43e6T^{2} \)
29 \( 1 + (-3.71e3 + 3.71e3i)T - 2.05e7iT^{2} \)
31 \( 1 - 4.55e3T + 2.86e7T^{2} \)
37 \( 1 + (7.33e3 + 7.33e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 857. iT - 1.15e8T^{2} \)
43 \( 1 + (-8.98e3 - 8.98e3i)T + 1.47e8iT^{2} \)
47 \( 1 - 2.00e4T + 2.29e8T^{2} \)
53 \( 1 + (-1.74e4 - 1.74e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (2.17e4 + 2.17e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (3.30e4 - 3.30e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (-2.10e3 + 2.10e3i)T - 1.35e9iT^{2} \)
71 \( 1 + 3.04e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.36e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.71e4T + 3.07e9T^{2} \)
83 \( 1 + (-2.14e3 + 2.14e3i)T - 3.93e9iT^{2} \)
89 \( 1 - 5.32e4iT - 5.58e9T^{2} \)
97 \( 1 + 5.74e3T + 8.58e9T^{2} \)
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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

−13.69206071946476750062336217097, −12.69330070493465896259571744898, −12.05273144230381805133034370894, −10.82515001631524417605502455155, −8.894045124573981986981771120815, −8.371163731271576841255813240496, −5.92387604252928129295462064975, −4.38528773481627308530714944715, −2.69264895783090894768981259673, −0.65730931715277615053696022149, 3.48746585152850681943295659421, 4.35970762674073796219356406577, 6.67330449010818659128071916140, 7.50161687210890271595928057522, 8.834203606466762345155079784357, 10.59800179626042600912251418883, 11.68714879741366348036604455255, 13.57421414475446588881194230510, 14.05948916893273970043648441688, 15.42019734418397094067182245617

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