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Properties

Label 2-48-12.11-c15-0-13
Degree $2$
Conductor $48$
Sign $-0.605 - 0.795i$
Analytic cond. $68.4928$
Root an. cond. $8.27604$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.29e3 + 3.01e3i)3-s + 7.60e4i·5-s + 1.82e6i·7-s + (−3.82e6 + 1.38e7i)9-s + 7.64e7·11-s + 2.54e8·13-s + (−2.29e8 + 1.74e8i)15-s + 7.38e8i·17-s + 5.32e8i·19-s + (−5.51e9 + 4.19e9i)21-s + 1.70e10·23-s + 2.47e10·25-s + (−5.04e10 + 2.01e10i)27-s + 1.64e11i·29-s − 1.19e11i·31-s + ⋯
L(s)  = 1  + (0.605 + 0.795i)3-s + 0.435i·5-s + 0.839i·7-s + (−0.266 + 0.963i)9-s + 1.18·11-s + 1.12·13-s + (−0.346 + 0.263i)15-s + 0.436i·17-s + 0.136i·19-s + (−0.668 + 0.508i)21-s + 1.04·23-s + 0.810·25-s + (−0.928 + 0.371i)27-s + 1.76i·29-s − 0.780i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(68.4928\)
Root analytic conductor: \(8.27604\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :15/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(8)\) \(\approx\) \(3.043649284\)
\(L(\frac12)\) \(\approx\) \(3.043649284\)
\(L(\frac{17}{2})\) 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 \( 1 + (-2.29e3 - 3.01e3i)T \)
good5 \( 1 - 7.60e4iT - 3.05e10T^{2} \)
7 \( 1 - 1.82e6iT - 4.74e12T^{2} \)
11 \( 1 - 7.64e7T + 4.17e15T^{2} \)
13 \( 1 - 2.54e8T + 5.11e16T^{2} \)
17 \( 1 - 7.38e8iT - 2.86e18T^{2} \)
19 \( 1 - 5.32e8iT - 1.51e19T^{2} \)
23 \( 1 - 1.70e10T + 2.66e20T^{2} \)
29 \( 1 - 1.64e11iT - 8.62e21T^{2} \)
31 \( 1 + 1.19e11iT - 2.34e22T^{2} \)
37 \( 1 + 1.78e11T + 3.33e23T^{2} \)
41 \( 1 + 5.29e11iT - 1.55e24T^{2} \)
43 \( 1 + 5.71e11iT - 3.17e24T^{2} \)
47 \( 1 + 1.37e12T + 1.20e25T^{2} \)
53 \( 1 + 1.30e13iT - 7.31e25T^{2} \)
59 \( 1 + 6.31e11T + 3.65e26T^{2} \)
61 \( 1 + 1.44e13T + 6.02e26T^{2} \)
67 \( 1 - 8.74e13iT - 2.46e27T^{2} \)
71 \( 1 + 8.26e13T + 5.87e27T^{2} \)
73 \( 1 - 6.67e13T + 8.90e27T^{2} \)
79 \( 1 - 1.93e14iT - 2.91e28T^{2} \)
83 \( 1 + 4.13e14T + 6.11e28T^{2} \)
89 \( 1 + 5.61e14iT - 1.74e29T^{2} \)
97 \( 1 + 3.75e14T + 6.33e29T^{2} \)
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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

−12.96691585088281271695471881800, −11.49371066896856498399271653476, −10.54366561422238416710079574618, −9.136911562735668118623506289126, −8.549698344058771585551128170460, −6.79822096231824439824239780855, −5.43066098635030943280914414504, −3.93002501915916307720937856840, −2.95061526568260118938856646253, −1.49427906368569442329413678384, 0.74614533500571646368450321670, 1.43009735753530188385247903199, 3.14604389155724359526172036267, 4.33138464077359477669754764547, 6.23755635808091637694483423312, 7.21775543341091262270398651033, 8.496409790432716710401090487074, 9.392280399850870046275211741681, 11.04986149095313060403935409169, 12.21044935010069398766339500970

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