↑

Properties

Label 2-1008-48.11-c1-0-31
Degree 22
Conductor 10081008
Sign 0.937+0.347i0.937 + 0.347i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.37 − 0.347i)2-s + (1.75 − 0.952i)4-s + (−0.111 + 0.111i)5-s + 7-s + (2.07 − 1.91i)8-s + (−0.114 + 0.191i)10-s + (3.61 + 3.61i)11-s + (−1.94 + 1.94i)13-s + (1.37 − 0.347i)14-s + (2.18 − 3.35i)16-s − 4.79i·17-s + (3.03 + 3.03i)19-s + (−0.0897 + 0.302i)20-s + (6.20 + 3.69i)22-s − 6.58i·23-s + ⋯
L(s)  = 1  + (0.969 − 0.245i)2-s + (0.879 − 0.476i)4-s + (−0.0498 + 0.0498i)5-s + 0.377·7-s + (0.735 − 0.677i)8-s + (−0.0360 + 0.0605i)10-s + (1.08 + 1.08i)11-s + (−0.539 + 0.539i)13-s + (0.366 − 0.0928i)14-s + (0.546 − 0.837i)16-s − 1.16i·17-s + (0.695 + 0.695i)19-s + (−0.0200 + 0.0675i)20-s + (1.32 + 0.788i)22-s − 1.37i·23-s + ⋯

Functional equation

Λ(s)=(1008s/2 ΓC(s) L(s)=((0.937+0.347i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1008s/2 ΓC(s+1/2) L(s)=((0.937+0.347i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10081008    =    24⋅32⋅72^{4} \cdot 3^{2} \cdot 7
Sign: 0.937+0.347i0.937 + 0.347i
Analytic conductor: 8.048928.04892
Root analytic conductor: 2.837062.83706
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1008(827,⋅)\chi_{1008} (827, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1008, ( :1/2), 0.937+0.347i)(2,\ 1008,\ (\ :1/2),\ 0.937 + 0.347i)

Particular Values

L(1)L(1) ≈\approx 3.2415422733.241542273
L(12)L(\frac12) ≈\approx 3.2415422733.241542273
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+(−1.37+0.347i)T 1 + (-1.37 + 0.347i)T
3 1 1
7 1−T 1 - T
good5 1+(0.111−0.111i)T−5iT2 1 + (0.111 - 0.111i)T - 5iT^{2}
11 1+(−3.61−3.61i)T+11iT2 1 + (-3.61 - 3.61i)T + 11iT^{2}
13 1+(1.94−1.94i)T−13iT2 1 + (1.94 - 1.94i)T - 13iT^{2}
17 1+4.79iT−17T2 1 + 4.79iT - 17T^{2}
19 1+(−3.03−3.03i)T+19iT2 1 + (-3.03 - 3.03i)T + 19iT^{2}
23 1+6.58iT−23T2 1 + 6.58iT - 23T^{2}
29 1+(−1.53−1.53i)T+29iT2 1 + (-1.53 - 1.53i)T + 29iT^{2}
31 1+3.26iT−31T2 1 + 3.26iT - 31T^{2}
37 1+(−1.05−1.05i)T+37iT2 1 + (-1.05 - 1.05i)T + 37iT^{2}
41 1+1.26T+41T2 1 + 1.26T + 41T^{2}
43 1+(−0.484+0.484i)T−43iT2 1 + (-0.484 + 0.484i)T - 43iT^{2}
47 1+11.2T+47T2 1 + 11.2T + 47T^{2}
53 1+(−4.00+4.00i)T−53iT2 1 + (-4.00 + 4.00i)T - 53iT^{2}
59 1+(7.61+7.61i)T+59iT2 1 + (7.61 + 7.61i)T + 59iT^{2}
61 1+(−5.44+5.44i)T−61iT2 1 + (-5.44 + 5.44i)T - 61iT^{2}
67 1+(0.897+0.897i)T+67iT2 1 + (0.897 + 0.897i)T + 67iT^{2}
71 1+2.83iT−71T2 1 + 2.83iT - 71T^{2}
73 1−15.7iT−73T2 1 - 15.7iT - 73T^{2}
79 1−15.4iT−79T2 1 - 15.4iT - 79T^{2}
83 1+(7.57−7.57i)T−83iT2 1 + (7.57 - 7.57i)T - 83iT^{2}
89 1−13.1T+89T2 1 - 13.1T + 89T^{2}
97 1+10.4T+97T2 1 + 10.4T + 97T^{2}
show more
show less
   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

−9.812507205296599539080500944229, −9.489364273118675161292556915110, −8.067529668863667604215525919610, −7.02847124278632703082117078351, −6.65465539180193336527595129732, −5.31518956503397593764839860770, −4.64618699035593809505245793976, −3.78155629384934498240161703692, −2.53800516235328119932742148681, −1.44791285681641722255307886889, 1.44519168101339132870353666056, 2.94039190550214937527690944843, 3.78942949687504357775809196919, 4.77328732952723708854015733287, 5.73215949074780299190583760278, 6.39326550860314895947563093863, 7.41278466759342619008203020164, 8.184756685403063109394023631639, 9.012559894220876219726968545168, 10.21253459170271093037905716886

Graph of the ZZ-function along the critical line