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Properties

Label 2-48-16.13-c3-0-3
Degree $2$
Conductor $48$
Sign $0.510 - 0.860i$
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  + (−1.92 − 2.07i)2-s + (2.12 + 2.12i)3-s + (−0.570 + 7.97i)4-s + (−7.29 + 7.29i)5-s + (0.302 − 8.47i)6-s + 22.1i·7-s + (17.6 − 14.1i)8-s + 8.99i·9-s + (29.1 + 1.04i)10-s + (8.24 − 8.24i)11-s + (−18.1 + 15.7i)12-s + (51.9 + 51.9i)13-s + (45.8 − 42.7i)14-s − 30.9·15-s + (−63.3 − 9.10i)16-s − 58.7·17-s + ⋯
L(s)  = 1  + (−0.681 − 0.731i)2-s + (0.408 + 0.408i)3-s + (−0.0713 + 0.997i)4-s + (−0.652 + 0.652i)5-s + (0.0205 − 0.576i)6-s + 1.19i·7-s + (0.778 − 0.627i)8-s + 0.333i·9-s + (0.921 + 0.0329i)10-s + (0.225 − 0.225i)11-s + (−0.436 + 0.378i)12-s + (1.10 + 1.10i)13-s + (0.875 − 0.815i)14-s − 0.532·15-s + (−0.989 − 0.142i)16-s − 0.837·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.860i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.510 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.510 - 0.860i$
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: \((2,\ 48,\ (\ :3/2),\ 0.510 - 0.860i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.798355 + 0.454654i\)
\(L(\frac12)\) \(\approx\) \(0.798355 + 0.454654i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.92 + 2.07i)T \)
3 \( 1 + (-2.12 - 2.12i)T \)
good5 \( 1 + (7.29 - 7.29i)T - 125iT^{2} \)
7 \( 1 - 22.1iT - 343T^{2} \)
11 \( 1 + (-8.24 + 8.24i)T - 1.33e3iT^{2} \)
13 \( 1 + (-51.9 - 51.9i)T + 2.19e3iT^{2} \)
17 \( 1 + 58.7T + 4.91e3T^{2} \)
19 \( 1 + (54.5 + 54.5i)T + 6.85e3iT^{2} \)
23 \( 1 + 117. iT - 1.21e4T^{2} \)
29 \( 1 + (-175. - 175. i)T + 2.43e4iT^{2} \)
31 \( 1 + 6.58T + 2.97e4T^{2} \)
37 \( 1 + (-265. + 265. i)T - 5.06e4iT^{2} \)
41 \( 1 - 98.7iT - 6.89e4T^{2} \)
43 \( 1 + (-347. + 347. i)T - 7.95e4iT^{2} \)
47 \( 1 + 141.T + 1.03e5T^{2} \)
53 \( 1 + (210. - 210. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-427. + 427. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-178. - 178. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-480. - 480. i)T + 3.00e5iT^{2} \)
71 \( 1 - 884. iT - 3.57e5T^{2} \)
73 \( 1 + 794. iT - 3.89e5T^{2} \)
79 \( 1 + 421.T + 4.93e5T^{2} \)
83 \( 1 + (167. + 167. i)T + 5.71e5iT^{2} \)
89 \( 1 - 664. iT - 7.04e5T^{2} \)
97 \( 1 + 1.08e3T + 9.12e5T^{2} \)
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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

−15.55590817692512096961694541241, −14.28136355898355303924534776898, −12.78506909337297744992985601191, −11.48190407364180775075747979479, −10.81424442384918648557130138513, −9.090029451013187823936069576703, −8.532756112205202863181465672666, −6.74146067920465247403293775318, −4.10489259130613309055192573450, −2.53329918797176210966640132807, 0.911332021036830210487955812288, 4.25361937003264696669497999183, 6.32047030299911131094372470059, 7.75694532232247241357358963237, 8.428419385939814932167284111149, 9.963994049277243860455130653173, 11.22839395012379550452489251251, 12.99177879254855796553550566548, 13.89538868537061058436654150981, 15.24568613772952653522484165332

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