Properties

Label 2-48-48.11-c3-0-9
Degree $2$
Conductor $48$
Sign $0.0960 - 0.995i$
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 + (5.07 − 1.13i)3-s + (−5.43 + 5.86i)4-s + (−11.2 + 11.2i)5-s + (8.68 + 11.8i)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 + (−20.9 + 35.9i)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 + (0.975 − 0.218i)3-s + (−0.679 + 0.733i)4-s + (−1.00 + 1.00i)5-s + (0.590 + 0.806i)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 + (−0.502 + 0.864i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0960 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0960 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.0960 - 0.995i$
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: \((2,\ 48,\ (\ :3/2),\ 0.0960 - 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.42111 + 1.29053i\)
\(L(\frac12)\) \(\approx\) \(1.42111 + 1.29053i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.07 + 1.13i)T \)
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

−15.16450493180902954097661752509, −14.46772166849411720954635320665, −13.70572525739049597907736287677, −12.11178373917899414960010154588, −10.85376063851485046466285783783, −8.700177779290641662756222528759, −7.85479146488833698649710659351, −6.99834432676598443049783402594, −4.74116660450628918308283819244, −3.16591804439138827737960065382, 1.72809329767637566374055517354, 4.02855920733033556038425290171, 4.87568048667022032470413173932, 8.079229289481370271792489482318, 8.587219399339664807108434498721, 10.24473387428427398804165280525, 11.53285746159668945799892871617, 12.52989284040409001587525263937, 13.68425139640393975082480997252, 14.83799940616768790375892509940

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