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Properties

Label 2-48-16.3-c2-0-0
Degree 22
Conductor 4848
Sign −0.933−0.357i-0.933 - 0.357i
Analytic cond. 1.307901.30790
Root an. cond. 1.143631.14363
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.84 + 0.777i)2-s + (−1.22 + 1.22i)3-s + (2.79 − 2.86i)4-s + (−4.78 + 4.78i)5-s + (1.30 − 3.20i)6-s − 10.3·7-s + (−2.91 + 7.45i)8-s − 2.99i·9-s + (5.09 − 12.5i)10-s + (−0.526 − 0.526i)11-s + (0.0930 + 6.92i)12-s + (17.2 + 17.2i)13-s + (19.0 − 8.03i)14-s − 11.7i·15-s + (−0.429 − 15.9i)16-s + 4.71·17-s + ⋯
L(s)  = 1  + (−0.921 + 0.388i)2-s + (−0.408 + 0.408i)3-s + (0.697 − 0.716i)4-s + (−0.957 + 0.957i)5-s + (0.217 − 0.534i)6-s − 1.47·7-s + (−0.364 + 0.931i)8-s − 0.333i·9-s + (0.509 − 1.25i)10-s + (−0.0478 − 0.0478i)11-s + (0.00775 + 0.577i)12-s + (1.32 + 1.32i)13-s + (1.35 − 0.573i)14-s − 0.781i·15-s + (−0.0268 − 0.999i)16-s + 0.277·17-s + ⋯

Functional equation

Λ(s)=(48s/2 ΓC(s) L(s)=((−0.933−0.357i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.357i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(48s/2 ΓC(s+1) L(s)=((−0.933−0.357i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.933 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4848    =    24⋅32^{4} \cdot 3
Sign: −0.933−0.357i-0.933 - 0.357i
Analytic conductor: 1.307901.30790
Root analytic conductor: 1.143631.14363
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ48(19,⋅)\chi_{48} (19, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 48, ( :1), −0.933−0.357i)(2,\ 48,\ (\ :1),\ -0.933 - 0.357i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.0629719+0.340402i0.0629719 + 0.340402i
L(12)L(\frac12) ≈\approx 0.0629719+0.340402i0.0629719 + 0.340402i
L(2)L(2) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(1.84−0.777i)T 1 + (1.84 - 0.777i)T
3 1+(1.22−1.22i)T 1 + (1.22 - 1.22i)T
good5 1+(4.78−4.78i)T−25iT2 1 + (4.78 - 4.78i)T - 25iT^{2}
7 1+10.3T+49T2 1 + 10.3T + 49T^{2}
11 1+(0.526+0.526i)T+121iT2 1 + (0.526 + 0.526i)T + 121iT^{2}
13 1+(−17.2−17.2i)T+169iT2 1 + (-17.2 - 17.2i)T + 169iT^{2}
17 1−4.71T+289T2 1 - 4.71T + 289T^{2}
19 1+(2.53−2.53i)T−361iT2 1 + (2.53 - 2.53i)T - 361iT^{2}
23 1+12.5T+529T2 1 + 12.5T + 529T^{2}
29 1+(2.19+2.19i)T+841iT2 1 + (2.19 + 2.19i)T + 841iT^{2}
31 1−28.0iT−961T2 1 - 28.0iT - 961T^{2}
37 1+(32.1−32.1i)T−1.36e3iT2 1 + (32.1 - 32.1i)T - 1.36e3iT^{2}
41 1+23.1iT−1.68e3T2 1 + 23.1iT - 1.68e3T^{2}
43 1+(−4.79−4.79i)T+1.84e3iT2 1 + (-4.79 - 4.79i)T + 1.84e3iT^{2}
47 1+39.0iT−2.20e3T2 1 + 39.0iT - 2.20e3T^{2}
53 1+(27.9−27.9i)T−2.80e3iT2 1 + (27.9 - 27.9i)T - 2.80e3iT^{2}
59 1+(−79.8−79.8i)T+3.48e3iT2 1 + (-79.8 - 79.8i)T + 3.48e3iT^{2}
61 1+(36.7+36.7i)T+3.72e3iT2 1 + (36.7 + 36.7i)T + 3.72e3iT^{2}
67 1+(10.9−10.9i)T−4.48e3iT2 1 + (10.9 - 10.9i)T - 4.48e3iT^{2}
71 1−52.6T+5.04e3T2 1 - 52.6T + 5.04e3T^{2}
73 1+67.8iT−5.32e3T2 1 + 67.8iT - 5.32e3T^{2}
79 1−56.4iT−6.24e3T2 1 - 56.4iT - 6.24e3T^{2}
83 1+(58.3−58.3i)T−6.88e3iT2 1 + (58.3 - 58.3i)T - 6.88e3iT^{2}
89 1−131.iT−7.92e3T2 1 - 131. iT - 7.92e3T^{2}
97 1−60.9T+9.40e3T2 1 - 60.9T + 9.40e3T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−16.00700335030649803580166157904, −15.35761418726261111186503233930, −13.96656529575482507407868602656, −12.00735153462705153357114514190, −10.99880154987976183564957314755, −10.01246272634389956576559926436, −8.734857522999047213730034458925, −7.02875048144783592790963465225, −6.22577571899850869857821276501, −3.54736934419977414921133792220, 0.49614870507904859657659736481, 3.54790274433153807732361457807, 6.12535704562413781966750438704, 7.69009386897323549059180946484, 8.738685452439597225442489337739, 10.12946265949644372215873927041, 11.40793802827014359211937876288, 12.62439923385948609598135588456, 13.02508946677102716289408745725, 15.78520197405580737027596421587

Graph of the ZZ-function along the critical line