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Properties

Label 2-48-16.11-c4-0-3
Degree $2$
Conductor $48$
Sign $-0.383 - 0.923i$
Analytic cond. $4.96175$
Root an. cond. $2.22750$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.82 + 2.82i)2-s + (3.67 + 3.67i)3-s + (−0.00384 − 15.9i)4-s + (21.6 + 21.6i)5-s + (−20.7 + 0.00249i)6-s + 15.5·7-s + (45.2 + 45.2i)8-s + 27i·9-s + (−122. + 0.0147i)10-s + (−48.9 + 48.9i)11-s + (58.7 − 58.8i)12-s + (−176. + 176. i)13-s + (−43.8 + 43.8i)14-s + 159. i·15-s + (−255. + 0.122i)16-s + 111.·17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.408 + 0.408i)3-s + (−0.000240 − 0.999i)4-s + (0.866 + 0.866i)5-s + (−0.577 + 6.93e−5i)6-s + 0.316·7-s + (0.707 + 0.706i)8-s + 0.333i·9-s + (−1.22 + 0.000147i)10-s + (−0.404 + 0.404i)11-s + (0.408 − 0.408i)12-s + (−1.04 + 1.04i)13-s + (−0.223 + 0.223i)14-s + 0.707i·15-s + (−0.999 + 0.000480i)16-s + 0.386·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.383 - 0.923i$
Analytic conductor: \(4.96175\)
Root analytic conductor: \(2.22750\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :2),\ -0.383 - 0.923i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.733410 + 1.09819i\)
\(L(\frac12)\) \(\approx\) \(0.733410 + 1.09819i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.82 - 2.82i)T \)
3 \( 1 + (-3.67 - 3.67i)T \)
good5 \( 1 + (-21.6 - 21.6i)T + 625iT^{2} \)
7 \( 1 - 15.5T + 2.40e3T^{2} \)
11 \( 1 + (48.9 - 48.9i)T - 1.46e4iT^{2} \)
13 \( 1 + (176. - 176. i)T - 2.85e4iT^{2} \)
17 \( 1 - 111.T + 8.35e4T^{2} \)
19 \( 1 + (-78.4 - 78.4i)T + 1.30e5iT^{2} \)
23 \( 1 - 745.T + 2.79e5T^{2} \)
29 \( 1 + (-32.1 + 32.1i)T - 7.07e5iT^{2} \)
31 \( 1 + 1.74e3iT - 9.23e5T^{2} \)
37 \( 1 + (-681. - 681. i)T + 1.87e6iT^{2} \)
41 \( 1 - 3.35e3iT - 2.82e6T^{2} \)
43 \( 1 + (-1.36e3 + 1.36e3i)T - 3.41e6iT^{2} \)
47 \( 1 + 3.77e3iT - 4.87e6T^{2} \)
53 \( 1 + (1.24e3 + 1.24e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (145. - 145. i)T - 1.21e7iT^{2} \)
61 \( 1 + (-4.34e3 + 4.34e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (-1.39e3 - 1.39e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 2.44e3T + 2.54e7T^{2} \)
73 \( 1 + 8.69e3iT - 2.83e7T^{2} \)
79 \( 1 + 5.55e3iT - 3.89e7T^{2} \)
83 \( 1 + (-3.39e3 - 3.39e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 7.37e3iT - 6.27e7T^{2} \)
97 \( 1 - 7.54e3T + 8.85e7T^{2} \)
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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

−14.98675230539021060681793667140, −14.59683868858877837830072830849, −13.46697140741790796408980446809, −11.31347691333526380484621650581, −10.04622037406518347249931390450, −9.416450117359775552086215858809, −7.78554736513213977352831358377, −6.59034183453814904929719544536, −4.98522468422212264040668836703, −2.25062344994760807802108332434, 1.06937480459965408601724222232, 2.78631254471133964097544665018, 5.18649450432425298326977991093, 7.40668204060540512928002561899, 8.611183647573420127929255784638, 9.589293787702901415690552001787, 10.82353641954402784164844068708, 12.46094457507442907328103002772, 12.99971528986873804150904160403, 14.26353614395151318098977107997

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