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Properties

Label 2-48-16.11-c4-0-14
Degree $2$
Conductor $48$
Sign $-0.559 + 0.828i$
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  + (3.43 − 2.05i)2-s + (−3.67 − 3.67i)3-s + (7.57 − 14.0i)4-s + (−15.8 − 15.8i)5-s + (−20.1 − 5.07i)6-s − 14.0·7-s + (−2.92 − 63.9i)8-s + 27i·9-s + (−86.8 − 21.8i)10-s + (37.6 − 37.6i)11-s + (−79.6 + 23.9i)12-s + (127. − 127. i)13-s + (−48.4 + 28.9i)14-s + 116. i·15-s + (−141. − 213. i)16-s − 81.0·17-s + ⋯
L(s)  = 1  + (0.858 − 0.513i)2-s + (−0.408 − 0.408i)3-s + (0.473 − 0.880i)4-s + (−0.633 − 0.633i)5-s + (−0.559 − 0.140i)6-s − 0.287·7-s + (−0.0456 − 0.998i)8-s + 0.333i·9-s + (−0.868 − 0.218i)10-s + (0.311 − 0.311i)11-s + (−0.552 + 0.166i)12-s + (0.753 − 0.753i)13-s + (−0.246 + 0.147i)14-s + 0.517i·15-s + (−0.551 − 0.833i)16-s − 0.280·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.559 + 0.828i$
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: \((2,\ 48,\ (\ :2),\ -0.559 + 0.828i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.867489 - 1.63187i\)
\(L(\frac12)\) \(\approx\) \(0.867489 - 1.63187i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.43 + 2.05i)T \)
3 \( 1 + (3.67 + 3.67i)T \)
good5 \( 1 + (15.8 + 15.8i)T + 625iT^{2} \)
7 \( 1 + 14.0T + 2.40e3T^{2} \)
11 \( 1 + (-37.6 + 37.6i)T - 1.46e4iT^{2} \)
13 \( 1 + (-127. + 127. i)T - 2.85e4iT^{2} \)
17 \( 1 + 81.0T + 8.35e4T^{2} \)
19 \( 1 + (-470. - 470. i)T + 1.30e5iT^{2} \)
23 \( 1 - 259.T + 2.79e5T^{2} \)
29 \( 1 + (-708. + 708. i)T - 7.07e5iT^{2} \)
31 \( 1 - 1.47e3iT - 9.23e5T^{2} \)
37 \( 1 + (-564. - 564. i)T + 1.87e6iT^{2} \)
41 \( 1 + 579. iT - 2.82e6T^{2} \)
43 \( 1 + (1.34e3 - 1.34e3i)T - 3.41e6iT^{2} \)
47 \( 1 + 2.48e3iT - 4.87e6T^{2} \)
53 \( 1 + (3.06e3 + 3.06e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (851. - 851. i)T - 1.21e7iT^{2} \)
61 \( 1 + (-3.22e3 + 3.22e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (-4.38e3 - 4.38e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 1.92e3T + 2.54e7T^{2} \)
73 \( 1 - 6.15e3iT - 2.83e7T^{2} \)
79 \( 1 + 3.81e3iT - 3.89e7T^{2} \)
83 \( 1 + (-6.19e3 - 6.19e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.50e4iT - 6.27e7T^{2} \)
97 \( 1 - 6.07e3T + 8.85e7T^{2} \)
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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

−14.23367688232272749007176272293, −13.14080216765115575251484133336, −12.22767744508712119933348312066, −11.38400028253374652352021585502, −10.03994776330420118671226322263, −8.208541071409670064034084398818, −6.48054358209750173110480252470, −5.14434879440377667146488345266, −3.48669916496907555110294075265, −1.01437186149531624657206646203, 3.28687343918843781807959637848, 4.67124543773803893527815265027, 6.35770215582782469876155208215, 7.38747308583880561415913013232, 9.175904741783019405882663333600, 11.08650962753287980782074776461, 11.69797184369541270517996952775, 13.13550201938227743226497592778, 14.30662528309638547573612527729, 15.41522581638572442994905636900

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