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Properties

Label 2-48-48.11-c3-0-4
Degree $2$
Conductor $48$
Sign $-0.998 + 0.0551i$
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  + (−0.763 + 2.72i)2-s + (−0.151 + 5.19i)3-s + (−6.83 − 4.16i)4-s + (−4.66 + 4.66i)5-s + (−14.0 − 4.37i)6-s + 0.405·7-s + (16.5 − 15.4i)8-s + (−26.9 − 1.56i)9-s + (−9.14 − 16.2i)10-s + (5.82 + 5.82i)11-s + (22.6 − 34.8i)12-s + (−35.2 + 35.2i)13-s + (−0.309 + 1.10i)14-s + (−23.5 − 24.9i)15-s + (29.3 + 56.8i)16-s + 49.3i·17-s + ⋯
L(s)  = 1  + (−0.270 + 0.962i)2-s + (−0.0290 + 0.999i)3-s + (−0.854 − 0.520i)4-s + (−0.417 + 0.417i)5-s + (−0.954 − 0.297i)6-s + 0.0219·7-s + (0.731 − 0.681i)8-s + (−0.998 − 0.0581i)9-s + (−0.289 − 0.514i)10-s + (0.159 + 0.159i)11-s + (0.544 − 0.838i)12-s + (−0.751 + 0.751i)13-s + (−0.00591 + 0.0210i)14-s + (−0.405 − 0.429i)15-s + (0.459 + 0.888i)16-s + 0.703i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0551i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.998 + 0.0551i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ -0.998 + 0.0551i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0226675 - 0.821479i\)
\(L(\frac12)\) \(\approx\) \(0.0226675 - 0.821479i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.763 - 2.72i)T \)
3 \( 1 + (0.151 - 5.19i)T \)
good5 \( 1 + (4.66 - 4.66i)T - 125iT^{2} \)
7 \( 1 - 0.405T + 343T^{2} \)
11 \( 1 + (-5.82 - 5.82i)T + 1.33e3iT^{2} \)
13 \( 1 + (35.2 - 35.2i)T - 2.19e3iT^{2} \)
17 \( 1 - 49.3iT - 4.91e3T^{2} \)
19 \( 1 + (-108. - 108. i)T + 6.85e3iT^{2} \)
23 \( 1 + 130. iT - 1.21e4T^{2} \)
29 \( 1 + (-172. - 172. i)T + 2.43e4iT^{2} \)
31 \( 1 + 36.1iT - 2.97e4T^{2} \)
37 \( 1 + (257. + 257. i)T + 5.06e4iT^{2} \)
41 \( 1 - 5.87T + 6.89e4T^{2} \)
43 \( 1 + (-170. + 170. i)T - 7.95e4iT^{2} \)
47 \( 1 - 181.T + 1.03e5T^{2} \)
53 \( 1 + (-148. + 148. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-567. - 567. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-481. + 481. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-296. - 296. i)T + 3.00e5iT^{2} \)
71 \( 1 + 533. iT - 3.57e5T^{2} \)
73 \( 1 - 178. iT - 3.89e5T^{2} \)
79 \( 1 - 528. iT - 4.93e5T^{2} \)
83 \( 1 + (713. - 713. i)T - 5.71e5iT^{2} \)
89 \( 1 + 204.T + 7.04e5T^{2} \)
97 \( 1 - 275.T + 9.12e5T^{2} \)
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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

−15.85864659136362454379914280404, −14.64926428428383618938081682565, −14.22270243530869243197567588420, −12.21437367338075107569854140043, −10.66156353774943636026199606780, −9.658304461511123360104786865231, −8.432217100182305397390946136204, −6.98811233538983009418831061966, −5.37784650979559802607034637808, −3.91169629840443603292543343358, 0.72907211890885063641490661631, 2.86927404444183221934782308157, 5.08344071072513140540336105183, 7.33284882631407605044475656299, 8.419509150364839189712496276303, 9.762166004350154983803619567183, 11.46706055767641111867017574641, 12.05767304257093760587933381684, 13.21861369596183085308426614253, 14.05489021666176984167030598515

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