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Properties

Label 2-48-48.11-c3-0-2
Degree $2$
Conductor $48$
Sign $-0.627 - 0.778i$
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 + (−4.66 + 2.28i)3-s + (−7.10 − 3.67i)4-s + (−11.5 + 11.5i)5-s + (3.15 + 14.3i)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 + (41.5 + 0.919i)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 + (−0.898 + 0.439i)3-s + (−0.888 − 0.459i)4-s + (−1.02 + 1.02i)5-s + (0.214 + 0.976i)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 + (0.999 + 0.0221i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.627 - 0.778i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.627 - 0.778i$
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: \((2,\ 48,\ (\ :3/2),\ -0.627 - 0.778i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0499064 + 0.104393i\)
\(L(\frac12)\) \(\approx\) \(0.0499064 + 0.104393i\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.66 - 2.28i)T \)
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

−15.40277710309908310613391460937, −14.52699822361292625509068476157, −12.92059675999175159861698812785, −11.78915962171040728231492375503, −10.87551918617867493397056539151, −10.33572889133508550494893593603, −8.393792058977718865400497124724, −6.43484714924371486767492387143, −4.70535813815580231908274319584, −3.23418625853571827518502948154, 0.090449985765555892133394440010, 4.51606467298014924222655000342, 5.43976708269892615487118006954, 7.32995628843620293457351939021, 7.982092507081764232578785122967, 9.763872417441568390308517939671, 11.68791472815630860873506295009, 12.58309240369293501754108411322, 13.34923049420647638400711160824, 15.21332260313639164127418939426

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