Properties

Label 2-48-12.11-c15-0-11
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

\(L(8)\) \(\approx\) \(1.342186579\)
\(L(\frac12)\) \(\approx\) \(1.342186579\)
\(L(\frac{17}{2})\) 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

−12.44304954841824158415787955110, −11.44157815080235524515692261281, −10.34026686630030112448783037183, −9.195988058212419284307564138144, −8.093353391676571613274937956617, −6.12814976085332948935557865351, −5.31225815278596181963730444402, −4.06007254625631983897053258357, −2.53386077561402317393481739727, −0.69318451926498322943669799507, 0.56202054419060885065662716996, 1.79862510225295728931495285638, 3.35518160707461158345502777291, 5.04924366922779471289801340004, 6.33761478187108456681660432742, 7.33728038281080805144677567052, 8.424771371638327941002219731326, 10.51067973459649017269942486741, 10.89406529731839947193598405522, 12.41646050013880727105739775177

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