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Properties

Label 2-48-16.5-c3-0-7
Degree $2$
Conductor $48$
Sign $0.944 - 0.328i$
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  + (2.59 + 1.12i)2-s + (2.12 − 2.12i)3-s + (5.49 + 5.81i)4-s + (0.706 + 0.706i)5-s + (7.88 − 3.13i)6-s − 4.44i·7-s + (7.74 + 21.2i)8-s − 8.99i·9-s + (1.04 + 2.62i)10-s + (−17.7 − 17.7i)11-s + (23.9 + 0.695i)12-s + (−17.7 + 17.7i)13-s + (4.97 − 11.5i)14-s + 2.99·15-s + (−3.70 + 63.8i)16-s − 105.·17-s + ⋯
L(s)  = 1  + (0.918 + 0.396i)2-s + (0.408 − 0.408i)3-s + (0.686 + 0.727i)4-s + (0.0631 + 0.0631i)5-s + (0.536 − 0.213i)6-s − 0.239i·7-s + (0.342 + 0.939i)8-s − 0.333i·9-s + (0.0330 + 0.0830i)10-s + (−0.485 − 0.485i)11-s + (0.577 + 0.0167i)12-s + (−0.377 + 0.377i)13-s + (0.0949 − 0.220i)14-s + 0.0516·15-s + (−0.0579 + 0.998i)16-s − 1.50·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.944 - 0.328i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ 0.944 - 0.328i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.32497 + 0.392833i\)
\(L(\frac12)\) \(\approx\) \(2.32497 + 0.392833i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.59 - 1.12i)T \)
3 \( 1 + (-2.12 + 2.12i)T \)
good5 \( 1 + (-0.706 - 0.706i)T + 125iT^{2} \)
7 \( 1 + 4.44iT - 343T^{2} \)
11 \( 1 + (17.7 + 17.7i)T + 1.33e3iT^{2} \)
13 \( 1 + (17.7 - 17.7i)T - 2.19e3iT^{2} \)
17 \( 1 + 105.T + 4.91e3T^{2} \)
19 \( 1 + (-40.2 + 40.2i)T - 6.85e3iT^{2} \)
23 \( 1 + 42.9iT - 1.21e4T^{2} \)
29 \( 1 + (185. - 185. i)T - 2.43e4iT^{2} \)
31 \( 1 - 291.T + 2.97e4T^{2} \)
37 \( 1 + (-151. - 151. i)T + 5.06e4iT^{2} \)
41 \( 1 - 60.3iT - 6.89e4T^{2} \)
43 \( 1 + (-120. - 120. i)T + 7.95e4iT^{2} \)
47 \( 1 - 500.T + 1.03e5T^{2} \)
53 \( 1 + (-192. - 192. i)T + 1.48e5iT^{2} \)
59 \( 1 + (500. + 500. i)T + 2.05e5iT^{2} \)
61 \( 1 + (166. - 166. i)T - 2.26e5iT^{2} \)
67 \( 1 + (575. - 575. i)T - 3.00e5iT^{2} \)
71 \( 1 + 457. iT - 3.57e5T^{2} \)
73 \( 1 + 1.08e3iT - 3.89e5T^{2} \)
79 \( 1 + 544.T + 4.93e5T^{2} \)
83 \( 1 + (-48.6 + 48.6i)T - 5.71e5iT^{2} \)
89 \( 1 - 43.6iT - 7.04e5T^{2} \)
97 \( 1 - 690.T + 9.12e5T^{2} \)
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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

−15.08858613993940191026481468835, −13.91173226431386540327336023578, −13.26388211121008955122671089934, −12.03516324172777026864344790885, −10.79032379884504446155462014968, −8.816288810806417876445963026812, −7.47363672214870598765253303267, −6.30147535932604432194332490848, −4.52860578003130304019176174588, −2.65758029675962945218904991401, 2.45512957280958678416242270630, 4.23960285823798504408022646404, 5.65481572924961537547830576869, 7.45250740426255156476437630907, 9.323564096826908413187650488558, 10.47965489213512710140943882856, 11.69889023470981999133412715891, 12.97990724372861499935272648990, 13.83648912677501756169903846027, 15.23118613468616385578419589518

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