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Properties

Label 2-1008-48.11-c1-0-6
Degree 22
Conductor 10081008
Sign 0.0154−0.999i0.0154 - 0.999i
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  + (−1.37 + 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)=((0.0154−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0154 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1008s/2 ΓC(s+1/2) L(s)=((0.0154−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0154 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10081008    =    24⋅32⋅72^{4} \cdot 3^{2} \cdot 7
Sign: 0.0154−0.999i0.0154 - 0.999i
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), 0.0154−0.999i)(2,\ 1008,\ (\ :1/2),\ 0.0154 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.77448877030.7744887703
L(12)L(\frac12) ≈\approx 0.77448877030.7744887703
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}
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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

−10.02737834158352849801042771844, −9.359360246011536244632598431928, −8.399552043671186465016374567640, −7.81770338149171083192405208085, −7.09588859333816108986826048746, −5.74832322088696928189830611033, −5.48219078375758223821830742236, −3.79330930167834979637158943131, −2.53688541688148165837219782932, −1.30596188196439714017395950010, 0.50709535867164819431240487624, 2.24890995147008382145191156539, 2.88216046519340911008301673706, 4.54259401650770868063519100515, 5.38371051337148325894580992278, 6.76764423801677947548310608416, 7.40448190731826674014299681753, 8.072478502729549054819099771362, 9.015604637665562922026068494919, 9.835446798491007958656056804135

Graph of the ZZ-function along the critical line