Properties

Label 2-48-3.2-c24-0-28
Degree $2$
Conductor $48$
Sign $0.507 + 0.861i$
Analytic cond. $175.184$
Root an. cond. $13.2357$
Motivic weight $24$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.69e5 − 4.58e5i)3-s + 4.56e8i·5-s − 1.81e9·7-s + (−1.37e11 + 2.46e11i)9-s − 3.36e12i·11-s + 3.10e13·13-s + (2.09e14 − 1.23e14i)15-s − 1.14e14i·17-s − 9.25e14·19-s + (4.90e14 + 8.33e14i)21-s + 2.51e16i·23-s − 1.48e17·25-s + (1.50e17 − 3.76e15i)27-s − 5.38e17i·29-s − 4.67e17·31-s + ⋯
L(s)  = 1  + (−0.507 − 0.861i)3-s + 1.87i·5-s − 0.131·7-s + (−0.485 + 0.874i)9-s − 1.07i·11-s + 1.33·13-s + (1.61 − 0.948i)15-s − 0.196i·17-s − 0.418·19-s + (0.0666 + 0.113i)21-s + 1.14i·23-s − 2.49·25-s + (0.999 − 0.0250i)27-s − 1.52i·29-s − 0.593·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.507 + 0.861i$
Analytic conductor: \(175.184\)
Root analytic conductor: \(13.2357\)
Motivic weight: \(24\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :12),\ 0.507 + 0.861i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(1.197481737\)
\(L(\frac12)\) \(\approx\) \(1.197481737\)
\(L(13)\) 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 \( 1 + (2.69e5 + 4.58e5i)T \)
good5 \( 1 - 4.56e8iT - 5.96e16T^{2} \)
7 \( 1 + 1.81e9T + 1.91e20T^{2} \)
11 \( 1 + 3.36e12iT - 9.84e24T^{2} \)
13 \( 1 - 3.10e13T + 5.42e26T^{2} \)
17 \( 1 + 1.14e14iT - 3.39e29T^{2} \)
19 \( 1 + 9.25e14T + 4.89e30T^{2} \)
23 \( 1 - 2.51e16iT - 4.80e32T^{2} \)
29 \( 1 + 5.38e17iT - 1.25e35T^{2} \)
31 \( 1 + 4.67e17T + 6.20e35T^{2} \)
37 \( 1 - 3.11e18T + 4.33e37T^{2} \)
41 \( 1 - 1.15e19iT - 5.09e38T^{2} \)
43 \( 1 + 5.41e19T + 1.59e39T^{2} \)
47 \( 1 + 1.41e20iT - 1.35e40T^{2} \)
53 \( 1 - 5.85e20iT - 2.41e41T^{2} \)
59 \( 1 - 1.22e21iT - 3.16e42T^{2} \)
61 \( 1 - 3.16e21T + 7.04e42T^{2} \)
67 \( 1 - 2.88e21T + 6.69e43T^{2} \)
71 \( 1 - 5.86e21iT - 2.69e44T^{2} \)
73 \( 1 + 4.89e21T + 5.24e44T^{2} \)
79 \( 1 + 4.71e22T + 3.49e45T^{2} \)
83 \( 1 + 1.25e23iT - 1.14e46T^{2} \)
89 \( 1 - 4.49e22iT - 6.10e46T^{2} \)
97 \( 1 + 2.08e23T + 4.81e47T^{2} \)
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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

−11.24544878025329614495456171741, −10.11748275451616478644044187104, −8.349048937086331948821108009367, −7.30819761845543113499550700724, −6.32662643160126341067723328457, −5.80528527030862643255250539413, −3.68561171634903034207413994091, −2.81739129759206069714673574120, −1.66820436529028736727679068003, −0.32694692915892374441726458299, 0.74170723258236286961662335591, 1.71897674192431920201505387069, 3.67305587964339001914624014351, 4.57381767390649318716991740325, 5.27825125689637153075945914094, 6.46831152375812830144358802189, 8.343861692117055455104106189875, 9.026730092020329481259385876818, 10.02104930237864421089629073757, 11.24272780128459051797167675290

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