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Properties

Label 2-48-16.13-c3-0-10
Degree $2$
Conductor $48$
Sign $-0.286 + 0.958i$
Analytic cond. $2.83209$
Root an. cond. $1.68288$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.94 − 2.05i)2-s + (−2.12 − 2.12i)3-s + (−0.408 − 7.98i)4-s + (2.24 − 2.24i)5-s + (−8.48 + 0.216i)6-s − 9.00i·7-s + (−17.1 − 14.7i)8-s + 8.99i·9-s + (−0.229 − 8.96i)10-s + (11.0 − 11.0i)11-s + (−16.0 + 17.8i)12-s + (54.5 + 54.5i)13-s + (−18.4 − 17.5i)14-s − 9.51·15-s + (−63.6 + 6.53i)16-s + 44.0·17-s + ⋯
L(s)  = 1  + (0.688 − 0.724i)2-s + (−0.408 − 0.408i)3-s + (−0.0510 − 0.998i)4-s + (0.200 − 0.200i)5-s + (−0.577 + 0.0147i)6-s − 0.486i·7-s + (−0.759 − 0.650i)8-s + 0.333i·9-s + (−0.00724 − 0.283i)10-s + (0.302 − 0.302i)11-s + (−0.386 + 0.428i)12-s + (1.16 + 1.16i)13-s + (−0.352 − 0.334i)14-s − 0.163·15-s + (−0.994 + 0.102i)16-s + 0.627·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.286 + 0.958i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ -0.286 + 0.958i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.01123 - 1.35773i\)
\(L(\frac12)\) \(\approx\) \(1.01123 - 1.35773i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.94 + 2.05i)T \)
3 \( 1 + (2.12 + 2.12i)T \)
good5 \( 1 + (-2.24 + 2.24i)T - 125iT^{2} \)
7 \( 1 + 9.00iT - 343T^{2} \)
11 \( 1 + (-11.0 + 11.0i)T - 1.33e3iT^{2} \)
13 \( 1 + (-54.5 - 54.5i)T + 2.19e3iT^{2} \)
17 \( 1 - 44.0T + 4.91e3T^{2} \)
19 \( 1 + (-49.9 - 49.9i)T + 6.85e3iT^{2} \)
23 \( 1 + 117. iT - 1.21e4T^{2} \)
29 \( 1 + (-40.6 - 40.6i)T + 2.43e4iT^{2} \)
31 \( 1 + 196.T + 2.97e4T^{2} \)
37 \( 1 + (248. - 248. i)T - 5.06e4iT^{2} \)
41 \( 1 + 457. iT - 6.89e4T^{2} \)
43 \( 1 + (-204. + 204. i)T - 7.95e4iT^{2} \)
47 \( 1 + 390.T + 1.03e5T^{2} \)
53 \( 1 + (138. - 138. i)T - 1.48e5iT^{2} \)
59 \( 1 + (263. - 263. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-29.1 - 29.1i)T + 2.26e5iT^{2} \)
67 \( 1 + (-508. - 508. i)T + 3.00e5iT^{2} \)
71 \( 1 - 788. iT - 3.57e5T^{2} \)
73 \( 1 - 92.2iT - 3.89e5T^{2} \)
79 \( 1 + 174.T + 4.93e5T^{2} \)
83 \( 1 + (-914. - 914. i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.45e3iT - 7.04e5T^{2} \)
97 \( 1 + 229.T + 9.12e5T^{2} \)
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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

−14.30265357571199789441564558077, −13.65177477498583057011143310910, −12.46922284560765949372343079375, −11.45668563753943357228887468299, −10.42749392678107716724305114305, −8.937290976993414032173219287104, −6.83595206513864537891308879022, −5.52922536228360921459003248683, −3.81008102065789634802619550318, −1.39464145672567039113512084883, 3.41698139568883987901826542814, 5.22689901345200853508595048627, 6.26961358005358694430456440400, 7.88829325974296723628293779497, 9.360242629972588664899270521275, 11.00085223287496421810059280079, 12.19886447594960233228278982510, 13.31874444102518935172405268532, 14.54113032409716800675042627098, 15.55630122657928266305738045752

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