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Properties

Label 2-48-48.11-c3-0-10
Degree $2$
Conductor $48$
Sign $0.859 + 0.511i$
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  + (−1.13 − 2.59i)2-s + (−1.13 + 5.07i)3-s + (−5.43 + 5.86i)4-s + (11.2 − 11.2i)5-s + (14.4 − 2.79i)6-s + 30.2·7-s + (21.3 + 7.45i)8-s + (−24.4 − 11.5i)9-s + (−41.9 − 16.4i)10-s + (14.9 + 14.9i)11-s + (−23.5 − 34.2i)12-s + (10.4 − 10.4i)13-s + (−34.1 − 78.3i)14-s + (44.3 + 69.9i)15-s + (−4.87 − 63.8i)16-s + 69.5i·17-s + ⋯
L(s)  = 1  + (−0.400 − 0.916i)2-s + (−0.218 + 0.975i)3-s + (−0.679 + 0.733i)4-s + (1.00 − 1.00i)5-s + (0.981 − 0.190i)6-s + 1.63·7-s + (0.944 + 0.329i)8-s + (−0.904 − 0.426i)9-s + (−1.32 − 0.520i)10-s + (0.410 + 0.410i)11-s + (−0.567 − 0.823i)12-s + (0.223 − 0.223i)13-s + (−0.652 − 1.49i)14-s + (0.763 + 1.20i)15-s + (−0.0761 − 0.997i)16-s + 0.992i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.859 + 0.511i$
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: \((2,\ 48,\ (\ :3/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.21482 - 0.334079i\)
\(L(\frac12)\) \(\approx\) \(1.21482 - 0.334079i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 + 2.59i)T \)
3 \( 1 + (1.13 - 5.07i)T \)
good5 \( 1 + (-11.2 + 11.2i)T - 125iT^{2} \)
7 \( 1 - 30.2T + 343T^{2} \)
11 \( 1 + (-14.9 - 14.9i)T + 1.33e3iT^{2} \)
13 \( 1 + (-10.4 + 10.4i)T - 2.19e3iT^{2} \)
17 \( 1 - 69.5iT - 4.91e3T^{2} \)
19 \( 1 + (36.4 + 36.4i)T + 6.85e3iT^{2} \)
23 \( 1 - 1.28iT - 1.21e4T^{2} \)
29 \( 1 + (119. + 119. i)T + 2.43e4iT^{2} \)
31 \( 1 - 172. iT - 2.97e4T^{2} \)
37 \( 1 + (235. + 235. i)T + 5.06e4iT^{2} \)
41 \( 1 - 19.9T + 6.89e4T^{2} \)
43 \( 1 + (294. - 294. i)T - 7.95e4iT^{2} \)
47 \( 1 - 69.5T + 1.03e5T^{2} \)
53 \( 1 + (-122. + 122. i)T - 1.48e5iT^{2} \)
59 \( 1 + (63.2 + 63.2i)T + 2.05e5iT^{2} \)
61 \( 1 + (352. - 352. i)T - 2.26e5iT^{2} \)
67 \( 1 + (228. + 228. i)T + 3.00e5iT^{2} \)
71 \( 1 - 524. iT - 3.57e5T^{2} \)
73 \( 1 + 578. iT - 3.89e5T^{2} \)
79 \( 1 - 745. iT - 4.93e5T^{2} \)
83 \( 1 + (286. - 286. i)T - 5.71e5iT^{2} \)
89 \( 1 - 203.T + 7.04e5T^{2} \)
97 \( 1 + 39.4T + 9.12e5T^{2} \)
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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

−14.95639044397610642125783611285, −13.83863608411236994882819895370, −12.50881735780693181929328129935, −11.31169573666037293706591515706, −10.34318624049976894331761366211, −9.141816392145901855833416883561, −8.335346053817569719440851078123, −5.35228053756784446301687926499, −4.31393340294875733313807628713, −1.66942740755141912364729858962, 1.72949027595329913754803695554, 5.31671224145831232495830192992, 6.49294667224135447489591291320, 7.58848671797927534355895701225, 8.831172603870245264726081108656, 10.53358671016184651353521984723, 11.56616176621810753675960491347, 13.58590597214720767871640006432, 14.15002179148780975988644518309, 14.93096682787437371498675337026

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