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Properties

Label 2-48-16.3-c6-0-13
Degree $2$
Conductor $48$
Sign $0.944 - 0.328i$
Analytic cond. $11.0425$
Root an. cond. $3.32304$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.82 + 1.67i)2-s + (−11.0 + 11.0i)3-s + (58.3 + 26.1i)4-s + (128. − 128. i)5-s + (−104. + 67.7i)6-s + 76.4·7-s + (412. + 302. i)8-s − 242. i·9-s + (1.21e3 − 788. i)10-s + (−565. − 565. i)11-s + (−932. + 354. i)12-s + (2.77e3 + 2.77e3i)13-s + (597. + 127. i)14-s + 2.82e3i·15-s + (2.72e3 + 3.05e3i)16-s + 5.24e3·17-s + ⋯
L(s)  = 1  + (0.977 + 0.209i)2-s + (−0.408 + 0.408i)3-s + (0.912 + 0.409i)4-s + (1.02 − 1.02i)5-s + (−0.484 + 0.313i)6-s + 0.222·7-s + (0.806 + 0.591i)8-s − 0.333i·9-s + (1.21 − 0.788i)10-s + (−0.425 − 0.425i)11-s + (−0.539 + 0.205i)12-s + (1.26 + 1.26i)13-s + (0.217 + 0.0466i)14-s + 0.838i·15-s + (0.664 + 0.746i)16-s + 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.944 - 0.328i$
Analytic conductor: \(11.0425\)
Root analytic conductor: \(3.32304\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3),\ 0.944 - 0.328i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.36285 + 0.568182i\)
\(L(\frac12)\) \(\approx\) \(3.36285 + 0.568182i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.82 - 1.67i)T \)
3 \( 1 + (11.0 - 11.0i)T \)
good5 \( 1 + (-128. + 128. i)T - 1.56e4iT^{2} \)
7 \( 1 - 76.4T + 1.17e5T^{2} \)
11 \( 1 + (565. + 565. i)T + 1.77e6iT^{2} \)
13 \( 1 + (-2.77e3 - 2.77e3i)T + 4.82e6iT^{2} \)
17 \( 1 - 5.24e3T + 2.41e7T^{2} \)
19 \( 1 + (46.8 - 46.8i)T - 4.70e7iT^{2} \)
23 \( 1 + 1.95e4T + 1.48e8T^{2} \)
29 \( 1 + (1.64e4 + 1.64e4i)T + 5.94e8iT^{2} \)
31 \( 1 + 2.40e4iT - 8.87e8T^{2} \)
37 \( 1 + (3.81e4 - 3.81e4i)T - 2.56e9iT^{2} \)
41 \( 1 + 6.65e4iT - 4.75e9T^{2} \)
43 \( 1 + (2.49e4 + 2.49e4i)T + 6.32e9iT^{2} \)
47 \( 1 - 1.20e5iT - 1.07e10T^{2} \)
53 \( 1 + (-9.27e4 + 9.27e4i)T - 2.21e10iT^{2} \)
59 \( 1 + (1.60e5 + 1.60e5i)T + 4.21e10iT^{2} \)
61 \( 1 + (1.78e5 + 1.78e5i)T + 5.15e10iT^{2} \)
67 \( 1 + (2.74e5 - 2.74e5i)T - 9.04e10iT^{2} \)
71 \( 1 - 6.31e5T + 1.28e11T^{2} \)
73 \( 1 - 4.70e5iT - 1.51e11T^{2} \)
79 \( 1 - 5.35e5iT - 2.43e11T^{2} \)
83 \( 1 + (2.84e5 - 2.84e5i)T - 3.26e11iT^{2} \)
89 \( 1 + 2.72e5iT - 4.96e11T^{2} \)
97 \( 1 - 4.99e5T + 8.32e11T^{2} \)
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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

−14.14059905322578637252922203595, −13.50079315065929836639429307389, −12.28580076589099977440072932224, −11.19927222270245617553112339249, −9.749004404152696672443267100068, −8.261254873232512880128587844561, −6.20495736137706916012646795052, −5.38344574868377511636333856865, −3.99200225237102113986170187502, −1.67407169828026000251077637640, 1.68184261510521590707209369473, 3.22981362282425074776435897301, 5.46067920560089716293151426938, 6.26552133242997279080441760506, 7.67173051728216893342325377405, 10.20184885657430248235196140143, 10.76877264541939493449955199010, 12.19627259444980599103949863587, 13.30422943933770352331601811938, 14.11983020404838263572164172564

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