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Properties

Label 2-1008-63.25-c1-0-19
Degree 22
Conductor 10081008
Sign 0.902+0.431i0.902 + 0.431i
Analytic cond. 8.048928.04892
Root an. cond. 2.837062.83706
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.73 + 0.0184i)3-s + (0.790 − 1.36i)5-s + (−2.57 − 0.601i)7-s + (2.99 − 0.0640i)9-s + (2.58 + 4.47i)11-s + (−0.681 − 1.18i)13-s + (−1.34 + 2.38i)15-s + (−2.30 + 3.99i)17-s + (−0.0321 − 0.0557i)19-s + (4.47 + 0.994i)21-s + (3.37 − 5.84i)23-s + (1.24 + 2.16i)25-s + (−5.19 + 0.166i)27-s + (4.70 − 8.15i)29-s + 2.66·31-s + ⋯
L(s)  = 1  + (−0.999 + 0.0106i)3-s + (0.353 − 0.612i)5-s + (−0.973 − 0.227i)7-s + (0.999 − 0.0213i)9-s + (0.779 + 1.35i)11-s + (−0.189 − 0.327i)13-s + (−0.347 + 0.616i)15-s + (−0.559 + 0.969i)17-s + (−0.00738 − 0.0127i)19-s + (0.976 + 0.216i)21-s + (0.703 − 1.21i)23-s + (0.249 + 0.432i)25-s + (−0.999 + 0.0320i)27-s + (0.874 − 1.51i)29-s + 0.478·31-s + ⋯

Functional equation

Λ(s)=(1008s/2 ΓC(s) L(s)=((0.902+0.431i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1008s/2 ΓC(s+1/2) L(s)=((0.902+0.431i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10081008    =    24⋅32⋅72^{4} \cdot 3^{2} \cdot 7
Sign: 0.902+0.431i0.902 + 0.431i
Analytic conductor: 8.048928.04892
Root analytic conductor: 2.837062.83706
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1008(529,⋅)\chi_{1008} (529, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1008, ( :1/2), 0.902+0.431i)(2,\ 1008,\ (\ :1/2),\ 0.902 + 0.431i)

Particular Values

L(1)L(1) ≈\approx 1.0884121291.088412129
L(12)L(\frac12) ≈\approx 1.0884121291.088412129
L(32)L(\frac{3}{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
3 1+(1.73−0.0184i)T 1 + (1.73 - 0.0184i)T
7 1+(2.57+0.601i)T 1 + (2.57 + 0.601i)T
good5 1+(−0.790+1.36i)T+(−2.5−4.33i)T2 1 + (-0.790 + 1.36i)T + (-2.5 - 4.33i)T^{2}
11 1+(−2.58−4.47i)T+(−5.5+9.52i)T2 1 + (-2.58 - 4.47i)T + (-5.5 + 9.52i)T^{2}
13 1+(0.681+1.18i)T+(−6.5+11.2i)T2 1 + (0.681 + 1.18i)T + (-6.5 + 11.2i)T^{2}
17 1+(2.30−3.99i)T+(−8.5−14.7i)T2 1 + (2.30 - 3.99i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.0321+0.0557i)T+(−9.5+16.4i)T2 1 + (0.0321 + 0.0557i)T + (-9.5 + 16.4i)T^{2}
23 1+(−3.37+5.84i)T+(−11.5−19.9i)T2 1 + (-3.37 + 5.84i)T + (-11.5 - 19.9i)T^{2}
29 1+(−4.70+8.15i)T+(−14.5−25.1i)T2 1 + (-4.70 + 8.15i)T + (-14.5 - 25.1i)T^{2}
31 1−2.66T+31T2 1 - 2.66T + 31T^{2}
37 1+(−0.880−1.52i)T+(−18.5+32.0i)T2 1 + (-0.880 - 1.52i)T + (-18.5 + 32.0i)T^{2}
41 1+(0.858+1.48i)T+(−20.5+35.5i)T2 1 + (0.858 + 1.48i)T + (-20.5 + 35.5i)T^{2}
43 1+(−5.12+8.86i)T+(−21.5−37.2i)T2 1 + (-5.12 + 8.86i)T + (-21.5 - 37.2i)T^{2}
47 1+5.20T+47T2 1 + 5.20T + 47T^{2}
53 1+(0.479−0.831i)T+(−26.5−45.8i)T2 1 + (0.479 - 0.831i)T + (-26.5 - 45.8i)T^{2}
59 1−9.33T+59T2 1 - 9.33T + 59T^{2}
61 1−14.3T+61T2 1 - 14.3T + 61T^{2}
67 1−12.4T+67T2 1 - 12.4T + 67T^{2}
71 1−4.49T+71T2 1 - 4.49T + 71T^{2}
73 1+(0.941−1.63i)T+(−36.5−63.2i)T2 1 + (0.941 - 1.63i)T + (-36.5 - 63.2i)T^{2}
79 1+6.53T+79T2 1 + 6.53T + 79T^{2}
83 1+(−5.08+8.81i)T+(−41.5−71.8i)T2 1 + (-5.08 + 8.81i)T + (-41.5 - 71.8i)T^{2}
89 1+(4.12+7.14i)T+(−44.5+77.0i)T2 1 + (4.12 + 7.14i)T + (-44.5 + 77.0i)T^{2}
97 1+(7.26−12.5i)T+(−48.5−84.0i)T2 1 + (7.26 - 12.5i)T + (-48.5 - 84.0i)T^{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

−9.977189117870323698851141178951, −9.334225332392388090437961821072, −8.312604269256964289706021756568, −6.93635700792709662094482723227, −6.63651327247862392936929314184, −5.63799641966201589040989315941, −4.63503644843900516965213782972, −3.96479453547305414648603663885, −2.21781919861293584544449991931, −0.794758937268402733944326751915, 0.943881466082071393086063855300, 2.73048240598811405082632134286, 3.68893396162874186594086168063, 4.99317958180120288338840615843, 5.87266391119547836949464398066, 6.70049516348071958728458025582, 6.95898102527743685785105458607, 8.500558419834844306407211515887, 9.447311441026437789235168750223, 9.960862643310531676635929750921

Graph of the ZZ-function along the critical line