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Properties

Label 2-48-16.11-c4-0-5
Degree $2$
Conductor $48$
Sign $0.969 + 0.244i$
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  + (−0.922 − 3.89i)2-s + (3.67 + 3.67i)3-s + (−14.2 + 7.18i)4-s + (17.5 + 17.5i)5-s + (10.9 − 17.6i)6-s + 50.9·7-s + (41.1 + 49.0i)8-s + 27i·9-s + (52.2 − 84.6i)10-s + (−2.46 + 2.46i)11-s + (−78.9 − 26.1i)12-s + (113. − 113. i)13-s + (−47.0 − 198. i)14-s + 129. i·15-s + (152. − 205. i)16-s − 89.8·17-s + ⋯
L(s)  = 1  + (−0.230 − 0.973i)2-s + (0.408 + 0.408i)3-s + (−0.893 + 0.448i)4-s + (0.703 + 0.703i)5-s + (0.303 − 0.491i)6-s + 1.04·7-s + (0.642 + 0.766i)8-s + 0.333i·9-s + (0.522 − 0.846i)10-s + (−0.0203 + 0.0203i)11-s + (−0.548 − 0.181i)12-s + (0.670 − 0.670i)13-s + (−0.240 − 1.01i)14-s + 0.574i·15-s + (0.597 − 0.802i)16-s − 0.310·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.969 + 0.244i$
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: \((2,\ 48,\ (\ :2),\ 0.969 + 0.244i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.68813 - 0.209718i\)
\(L(\frac12)\) \(\approx\) \(1.68813 - 0.209718i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.922 + 3.89i)T \)
3 \( 1 + (-3.67 - 3.67i)T \)
good5 \( 1 + (-17.5 - 17.5i)T + 625iT^{2} \)
7 \( 1 - 50.9T + 2.40e3T^{2} \)
11 \( 1 + (2.46 - 2.46i)T - 1.46e4iT^{2} \)
13 \( 1 + (-113. + 113. i)T - 2.85e4iT^{2} \)
17 \( 1 + 89.8T + 8.35e4T^{2} \)
19 \( 1 + (-355. - 355. i)T + 1.30e5iT^{2} \)
23 \( 1 + 381.T + 2.79e5T^{2} \)
29 \( 1 + (636. - 636. i)T - 7.07e5iT^{2} \)
31 \( 1 - 949. iT - 9.23e5T^{2} \)
37 \( 1 + (1.56e3 + 1.56e3i)T + 1.87e6iT^{2} \)
41 \( 1 + 2.26e3iT - 2.82e6T^{2} \)
43 \( 1 + (-2.58e3 + 2.58e3i)T - 3.41e6iT^{2} \)
47 \( 1 - 2.69e3iT - 4.87e6T^{2} \)
53 \( 1 + (3.73e3 + 3.73e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (215. - 215. i)T - 1.21e7iT^{2} \)
61 \( 1 + (-2.76e3 + 2.76e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (5.08e3 + 5.08e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 3.46e3T + 2.54e7T^{2} \)
73 \( 1 - 2.03e3iT - 2.83e7T^{2} \)
79 \( 1 + 2.99e3iT - 3.89e7T^{2} \)
83 \( 1 + (-6.01e3 - 6.01e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 8.58e3iT - 6.27e7T^{2} \)
97 \( 1 - 6.47e3T + 8.85e7T^{2} \)
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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

−14.31804334891801871690899913983, −13.97730121977795537807105457527, −12.41694953365444796308896564972, −10.96896938814687182800438709705, −10.32628909646037474010975148604, −9.003262686756672942486140275900, −7.77073053758473044909401854207, −5.38558783553141549325064029971, −3.56611064853542385832801707844, −1.88423367870335482406797892887, 1.42266495040833589094348932735, 4.61674463223470929363485063169, 6.00812063717674084868835593913, 7.56650533608702039920871769233, 8.686488594291273567092832089396, 9.602861082670764786212755970789, 11.46601977520635308448365610999, 13.24470792497740724272935443096, 13.79972922511771496120240301767, 14.91592479532093832837817553397

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