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Properties

Label 2-48-48.35-c3-0-3
Degree $2$
Conductor $48$
Sign $0.973 - 0.228i$
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  + (−0.668 − 2.74i)2-s + (2.28 + 4.66i)3-s + (−7.10 + 3.67i)4-s + (11.5 + 11.5i)5-s + (11.3 − 9.39i)6-s + 0.829·7-s + (14.8 + 17.0i)8-s + (−16.5 + 21.3i)9-s + (23.9 − 39.3i)10-s + (38.2 − 38.2i)11-s + (−33.3 − 24.7i)12-s + (−20.0 − 20.0i)13-s + (−0.554 − 2.27i)14-s + (−27.4 + 80.0i)15-s + (37.0 − 52.2i)16-s + 77.9i·17-s + ⋯
L(s)  = 1  + (−0.236 − 0.971i)2-s + (0.439 + 0.898i)3-s + (−0.888 + 0.459i)4-s + (1.02 + 1.02i)5-s + (0.769 − 0.639i)6-s + 0.0447·7-s + (0.656 + 0.754i)8-s + (−0.613 + 0.789i)9-s + (0.757 − 1.24i)10-s + (1.04 − 1.04i)11-s + (−0.802 − 0.596i)12-s + (−0.427 − 0.427i)13-s + (−0.0105 − 0.0435i)14-s + (−0.472 + 1.37i)15-s + (0.578 − 0.815i)16-s + 1.11i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.973 - 0.228i$
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: \((2,\ 48,\ (\ :3/2),\ 0.973 - 0.228i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.42544 + 0.165332i\)
\(L(\frac12)\) \(\approx\) \(1.42544 + 0.165332i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.668 + 2.74i)T \)
3 \( 1 + (-2.28 - 4.66i)T \)
good5 \( 1 + (-11.5 - 11.5i)T + 125iT^{2} \)
7 \( 1 - 0.829T + 343T^{2} \)
11 \( 1 + (-38.2 + 38.2i)T - 1.33e3iT^{2} \)
13 \( 1 + (20.0 + 20.0i)T + 2.19e3iT^{2} \)
17 \( 1 - 77.9iT - 4.91e3T^{2} \)
19 \( 1 + (37.6 - 37.6i)T - 6.85e3iT^{2} \)
23 \( 1 + 159. iT - 1.21e4T^{2} \)
29 \( 1 + (26.6 - 26.6i)T - 2.43e4iT^{2} \)
31 \( 1 + 226. iT - 2.97e4T^{2} \)
37 \( 1 + (-176. + 176. i)T - 5.06e4iT^{2} \)
41 \( 1 - 301.T + 6.89e4T^{2} \)
43 \( 1 + (96.1 + 96.1i)T + 7.95e4iT^{2} \)
47 \( 1 - 122.T + 1.03e5T^{2} \)
53 \( 1 + (138. + 138. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-168. + 168. i)T - 2.05e5iT^{2} \)
61 \( 1 + (178. + 178. i)T + 2.26e5iT^{2} \)
67 \( 1 + (233. - 233. i)T - 3.00e5iT^{2} \)
71 \( 1 - 668. iT - 3.57e5T^{2} \)
73 \( 1 - 1.21e3iT - 3.89e5T^{2} \)
79 \( 1 - 39.4iT - 4.93e5T^{2} \)
83 \( 1 + (-293. - 293. i)T + 5.71e5iT^{2} \)
89 \( 1 - 732.T + 7.04e5T^{2} \)
97 \( 1 - 451.T + 9.12e5T^{2} \)
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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

−14.62556448950172134443960977619, −14.29530318635695867540375247787, −12.96209602678313912333526667035, −11.19764115992398228312235382539, −10.44924013176363458479683456457, −9.514681164935035717095026691652, −8.322744406913096538657501386938, −5.97472388921191164291469278328, −3.91513059434977325445606935707, −2.46564565460772114746897219584, 1.47122674124574118529532102695, 4.85252296524469041836224969725, 6.37107247432593691375557978887, 7.51185687867226043300166105167, 9.127238000983373509441442601890, 9.481534984926406057066731758659, 12.06496633388125160063473786197, 13.18855485356862869819745998589, 13.96360765021891446249409255884, 14.95619005749241278953033052448

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