Properties

Label 2-48-12.11-c1-0-1
Degree $2$
Conductor $48$
Sign $0.866 + 0.5i$
Analytic cond. $0.383281$
Root an. cond. $0.619097$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·3-s + 3.46i·7-s − 2.99·9-s − 2·13-s − 3.46i·19-s + 5.99·21-s + 5·25-s + 5.19i·27-s − 10.3i·31-s − 10·37-s + 3.46i·39-s + 10.3i·43-s − 4.99·49-s − 5.99·57-s + 14·61-s + ⋯
L(s)  = 1  − 0.999i·3-s + 1.30i·7-s − 0.999·9-s − 0.554·13-s − 0.794i·19-s + 1.30·21-s + 25-s + 0.999i·27-s − 1.86i·31-s − 1.64·37-s + 0.554i·39-s + 1.58i·43-s − 0.714·49-s − 0.794·57-s + 1.79·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(0.383281\)
Root analytic conductor: \(0.619097\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1/2),\ 0.866 + 0.5i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.771843 - 0.206814i\)
\(L(\frac12)\) \(\approx\) \(0.771843 - 0.206814i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
good5 \( 1 - 5T^{2} \)
7 \( 1 - 3.46iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 10.3iT - 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 17.3iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−15.44375138708661754247537486481, −14.46956971764483229019315726323, −13.12925026034704855482017874281, −12.24056695896816489745402282108, −11.26412486568634322020379293105, −9.338892356501978764549936387737, −8.201099013122993350293745119564, −6.74418422965018797428766625394, −5.37448383935061293331068662521, −2.52494733590377164458054694339, 3.63537616080026223751227237238, 5.06935218801281721891351547462, 7.02451456717377461766522669630, 8.625402204712350869792689549676, 10.10407073322686381731412723868, 10.73331120502494611207295480445, 12.23659147913522161754220310176, 13.83864131223898957951710912710, 14.60504122510036768778392573017, 15.92369172058273067985032747780

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