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Properties

Label 2-48-16.3-c6-0-9
Degree $2$
Conductor $48$
Sign $0.866 + 0.500i$
Analytic cond. $11.0425$
Root an. cond. $3.32304$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.30 + 7.28i)2-s + (−11.0 + 11.0i)3-s + (−42.1 − 48.1i)4-s + (−98.7 + 98.7i)5-s + (−43.9 − 116. i)6-s − 218.·7-s + (489. − 148. i)8-s − 242. i·9-s + (−393. − 1.04e3i)10-s + (−24.6 − 24.6i)11-s + (995. + 65.2i)12-s + (−655. − 655. i)13-s + (721. − 1.59e3i)14-s − 2.17e3i·15-s + (−534. + 4.06e3i)16-s + 6.23e3·17-s + ⋯
L(s)  = 1  + (−0.412 + 0.910i)2-s + (−0.408 + 0.408i)3-s + (−0.659 − 0.751i)4-s + (−0.790 + 0.790i)5-s + (−0.203 − 0.540i)6-s − 0.636·7-s + (0.956 − 0.290i)8-s − 0.333i·9-s + (−0.393 − 1.04i)10-s + (−0.0185 − 0.0185i)11-s + (0.576 + 0.0377i)12-s + (−0.298 − 0.298i)13-s + (0.262 − 0.579i)14-s − 0.645i·15-s + (−0.130 + 0.991i)16-s + 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.500i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.866 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.866 + 0.500i$
Analytic conductor: \(11.0425\)
Root analytic conductor: \(3.32304\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3),\ 0.866 + 0.500i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.367974 - 0.0986085i\)
\(L(\frac12)\) \(\approx\) \(0.367974 - 0.0986085i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.30 - 7.28i)T \)
3 \( 1 + (11.0 - 11.0i)T \)
good5 \( 1 + (98.7 - 98.7i)T - 1.56e4iT^{2} \)
7 \( 1 + 218.T + 1.17e5T^{2} \)
11 \( 1 + (24.6 + 24.6i)T + 1.77e6iT^{2} \)
13 \( 1 + (655. + 655. i)T + 4.82e6iT^{2} \)
17 \( 1 - 6.23e3T + 2.41e7T^{2} \)
19 \( 1 + (-2.07e3 + 2.07e3i)T - 4.70e7iT^{2} \)
23 \( 1 - 8.16e3T + 1.48e8T^{2} \)
29 \( 1 + (1.96e4 + 1.96e4i)T + 5.94e8iT^{2} \)
31 \( 1 + 5.29e4iT - 8.87e8T^{2} \)
37 \( 1 + (6.46e4 - 6.46e4i)T - 2.56e9iT^{2} \)
41 \( 1 + 5.79e4iT - 4.75e9T^{2} \)
43 \( 1 + (4.80e4 + 4.80e4i)T + 6.32e9iT^{2} \)
47 \( 1 + 1.01e5iT - 1.07e10T^{2} \)
53 \( 1 + (-1.57e4 + 1.57e4i)T - 2.21e10iT^{2} \)
59 \( 1 + (7.38e4 + 7.38e4i)T + 4.21e10iT^{2} \)
61 \( 1 + (-1.90e5 - 1.90e5i)T + 5.15e10iT^{2} \)
67 \( 1 + (-1.48e5 + 1.48e5i)T - 9.04e10iT^{2} \)
71 \( 1 - 4.72e4T + 1.28e11T^{2} \)
73 \( 1 + 8.10e4iT - 1.51e11T^{2} \)
79 \( 1 + 5.34e5iT - 2.43e11T^{2} \)
83 \( 1 + (-6.37e5 + 6.37e5i)T - 3.26e11iT^{2} \)
89 \( 1 - 1.13e6iT - 4.96e11T^{2} \)
97 \( 1 + 5.18e5T + 8.32e11T^{2} \)
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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

−14.87379813326901812528955941957, −13.39987978975855574674607529609, −11.77976392954352610176585954047, −10.47922846712075882318489475502, −9.511585739473718073621112846355, −7.85558367586310453281353516226, −6.79616508679265454691938589316, −5.39694556954055860333875217166, −3.60609270218699016297179717175, −0.24490107349320867850693603990, 1.20722390542281832329089791940, 3.41012828350876149469869882117, 5.05988532397306820873968099065, 7.26357630646779581437382788170, 8.498428465419009935417171670786, 9.762585999391247534208327138254, 11.12240434940600809506051948206, 12.34835664926036915047539396608, 12.65479106553207024691501145188, 14.18947471432927582629912085956

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