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Properties

Label 2-48-48.35-c3-0-9
Degree $2$
Conductor $48$
Sign $0.732 + 0.681i$
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  + (−2.60 − 1.09i)2-s + (4.68 − 2.24i)3-s + (5.60 + 5.70i)4-s + (2.69 + 2.69i)5-s + (−14.6 + 0.719i)6-s + 10.6·7-s + (−8.38 − 21.0i)8-s + (16.9 − 21.0i)9-s + (−4.07 − 9.96i)10-s + (29.3 − 29.3i)11-s + (39.0 + 14.1i)12-s + (−7.80 − 7.80i)13-s + (−27.7 − 11.6i)14-s + (18.6 + 6.58i)15-s + (−1.10 + 63.9i)16-s − 13.2i·17-s + ⋯
L(s)  = 1  + (−0.922 − 0.386i)2-s + (0.902 − 0.431i)3-s + (0.700 + 0.713i)4-s + (0.240 + 0.240i)5-s + (−0.998 + 0.0489i)6-s + 0.574·7-s + (−0.370 − 0.928i)8-s + (0.627 − 0.778i)9-s + (−0.128 − 0.314i)10-s + (0.804 − 0.804i)11-s + (0.940 + 0.341i)12-s + (−0.166 − 0.166i)13-s + (−0.529 − 0.222i)14-s + (0.320 + 0.113i)15-s + (−0.0172 + 0.999i)16-s − 0.188i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.681i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.732 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.732 + 0.681i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ 0.732 + 0.681i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.20147 - 0.472499i\)
\(L(\frac12)\) \(\approx\) \(1.20147 - 0.472499i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.60 + 1.09i)T \)
3 \( 1 + (-4.68 + 2.24i)T \)
good5 \( 1 + (-2.69 - 2.69i)T + 125iT^{2} \)
7 \( 1 - 10.6T + 343T^{2} \)
11 \( 1 + (-29.3 + 29.3i)T - 1.33e3iT^{2} \)
13 \( 1 + (7.80 + 7.80i)T + 2.19e3iT^{2} \)
17 \( 1 + 13.2iT - 4.91e3T^{2} \)
19 \( 1 + (85.6 - 85.6i)T - 6.85e3iT^{2} \)
23 \( 1 - 166. iT - 1.21e4T^{2} \)
29 \( 1 + (58.7 - 58.7i)T - 2.43e4iT^{2} \)
31 \( 1 - 249. iT - 2.97e4T^{2} \)
37 \( 1 + (-174. + 174. i)T - 5.06e4iT^{2} \)
41 \( 1 + 469.T + 6.89e4T^{2} \)
43 \( 1 + (-86.4 - 86.4i)T + 7.95e4iT^{2} \)
47 \( 1 + 585.T + 1.03e5T^{2} \)
53 \( 1 + (-318. - 318. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-273. + 273. i)T - 2.05e5iT^{2} \)
61 \( 1 + (270. + 270. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-241. + 241. i)T - 3.00e5iT^{2} \)
71 \( 1 - 203. iT - 3.57e5T^{2} \)
73 \( 1 - 47.3iT - 3.89e5T^{2} \)
79 \( 1 + 160. iT - 4.93e5T^{2} \)
83 \( 1 + (-382. - 382. i)T + 5.71e5iT^{2} \)
89 \( 1 - 588.T + 7.04e5T^{2} \)
97 \( 1 + 172.T + 9.12e5T^{2} \)
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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

−14.93646686979651411187625146911, −13.94977329303884422453047124017, −12.58405192427644591529505548561, −11.41269869445366180152430895081, −10.04830972918130226836897674299, −8.830559762654521631903498463385, −7.912355676896812102140719878275, −6.52226558002929891501548773764, −3.47475951173816391912406043944, −1.66281761832468928289102389688, 2.03787196967896598845441683455, 4.66368196309372501203957653339, 6.77321229045906311712914896090, 8.174313887246404772208951052371, 9.137829789152023404926105115449, 10.12431769187257031702756871708, 11.45768028130142201836365635157, 13.23068180967347627710903115292, 14.84998739721233754364874740280, 14.96275229163376826665443404421

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