Properties

Label 2-48-12.11-c5-0-8
Degree $2$
Conductor $48$
Sign $-0.866 + 0.499i$
Analytic cond. $7.69842$
Root an. cond. $2.77460$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 15.5i·3-s − 107. i·7-s − 243·9-s − 1.20e3·13-s − 2.80e3i·19-s − 1.67e3·21-s + 3.12e3·25-s + 3.78e3i·27-s + 2.81e3i·31-s + 1.65e4·37-s + 1.87e4i·39-s − 2.40e4i·43-s + 5.27e3·49-s − 4.36e4·57-s − 3.86e4·61-s + ⋯
L(s)  = 1  − 0.999i·3-s − 0.828i·7-s − 9-s − 1.97·13-s − 1.78i·19-s − 0.828·21-s + 25-s + 1.00i·27-s + 0.526i·31-s + 1.98·37-s + 1.97i·39-s − 1.98i·43-s + 0.313·49-s − 1.78·57-s − 1.32·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.866 + 0.499i$
Analytic conductor: \(7.69842\)
Root analytic conductor: \(2.77460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :5/2),\ -0.866 + 0.499i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.273384 - 1.02028i\)
\(L(\frac12)\) \(\approx\) \(0.273384 - 1.02028i\)
\(L(\frac{7}{2})\) 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 \( 1 + 15.5iT \)
good5 \( 1 - 3.12e3T^{2} \)
7 \( 1 + 107. iT - 1.68e4T^{2} \)
11 \( 1 + 1.61e5T^{2} \)
13 \( 1 + 1.20e3T + 3.71e5T^{2} \)
17 \( 1 - 1.41e6T^{2} \)
19 \( 1 + 2.80e3iT - 2.47e6T^{2} \)
23 \( 1 + 6.43e6T^{2} \)
29 \( 1 - 2.05e7T^{2} \)
31 \( 1 - 2.81e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.65e4T + 6.93e7T^{2} \)
41 \( 1 - 1.15e8T^{2} \)
43 \( 1 + 2.40e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.29e8T^{2} \)
53 \( 1 - 4.18e8T^{2} \)
59 \( 1 + 7.14e8T^{2} \)
61 \( 1 + 3.86e4T + 8.44e8T^{2} \)
67 \( 1 + 6.43e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.80e9T^{2} \)
73 \( 1 - 1.45e3T + 2.07e9T^{2} \)
79 \( 1 - 4.68e4iT - 3.07e9T^{2} \)
83 \( 1 + 3.93e9T^{2} \)
89 \( 1 - 5.58e9T^{2} \)
97 \( 1 - 1.34e5T + 8.58e9T^{2} \)
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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

−14.03699098206157437608019050711, −13.02435106977880487577777503141, −12.00894012259824749626309218686, −10.73294575317049869647731396624, −9.202446376916645672987468336133, −7.58625326679362375573330017890, −6.81550699366384216420948206783, −4.90379891855357016209382653030, −2.58921108622822011621847214912, −0.52985752654061047898380275980, 2.70826022295042964647823495122, 4.57919247105348997594189761510, 5.86379323120899748946715646826, 7.891029212429343711132222885741, 9.325957207235366741403073429107, 10.16702957579116635747158192743, 11.62697941999381711530319783869, 12.60489670257376498889653323020, 14.54987820751536153950059612806, 14.88833208845833776686730933753

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