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Properties

Label 2-48-12.11-c3-0-5
Degree $2$
Conductor $48$
Sign $i$
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  − 5.19i·3-s − 31.1i·7-s − 27·9-s + 70·13-s + 155. i·19-s − 162·21-s + 125·25-s + 140. i·27-s − 155. i·31-s + 110·37-s − 363. i·39-s − 218. i·43-s − 629·49-s + 810·57-s + 182·61-s + ⋯
L(s)  = 1  − 0.999i·3-s − 1.68i·7-s − 9-s + 1.49·13-s + 1.88i·19-s − 1.68·21-s + 25-s + 1.00i·27-s − 0.903i·31-s + 0.488·37-s − 1.49i·39-s − 0.773i·43-s − 1.83·49-s + 1.88·57-s + 0.382·61-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $i$
Analytic conductor: \(2.83209\)
Root analytic conductor: \(1.68288\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3/2),\ i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.918715 - 0.918715i\)
\(L(\frac12)\) \(\approx\) \(0.918715 - 0.918715i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 5.19iT \)
good5 \( 1 - 125T^{2} \)
7 \( 1 + 31.1iT - 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 - 70T + 2.19e3T^{2} \)
17 \( 1 - 4.91e3T^{2} \)
19 \( 1 - 155. iT - 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + 155. iT - 2.97e4T^{2} \)
37 \( 1 - 110T + 5.06e4T^{2} \)
41 \( 1 - 6.89e4T^{2} \)
43 \( 1 + 218. iT - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 - 182T + 2.26e5T^{2} \)
67 \( 1 - 654. iT - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + 1.19e3T + 3.89e5T^{2} \)
79 \( 1 - 1.09e3iT - 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 - 7.04e5T^{2} \)
97 \( 1 - 1.33e3T + 9.12e5T^{2} \)
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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

−14.48397440827174773566266841978, −13.65871104122632564860206170661, −12.81149062584364792423133025021, −11.35136314636757206169758619459, −10.30643596198444683353951289091, −8.404689753214811157991635364118, −7.32711469925858239422710865299, −6.08328547140778137441328859160, −3.77976963549535500715775465020, −1.16406634444353320136248559263, 2.95679205756170346011661390547, 4.92537639830739650212114525279, 6.20313830914802857444893198694, 8.619447060864028695969062358540, 9.173692047359594035107973941111, 10.82246833981666451061679104804, 11.72123289506996842259428660147, 13.16639203226429940120011308065, 14.64698277918031551293222793826, 15.57247524311043940173259685686

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