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Properties

Label 2-1008-21.11-c0-0-1
Degree 22
Conductor 10081008
Sign −0.851+0.524i-0.851 + 0.524i
Analytic cond. 0.5030570.503057
Root an. cond. 0.7092650.709265
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 0.707i)5-s + (−0.5 − 0.866i)7-s + (−1.22 − 0.707i)11-s − 13-s + (−0.5 − 0.866i)19-s + (0.499 − 0.866i)25-s + (−0.5 + 0.866i)31-s + (1.22 + 0.707i)35-s + (−0.5 − 0.866i)37-s − 1.41i·41-s + 43-s + (−1.22 + 0.707i)47-s + (−0.499 + 0.866i)49-s + 2·55-s + (1.22 − 0.707i)65-s + ⋯
L(s)  = 1  + (−1.22 + 0.707i)5-s + (−0.5 − 0.866i)7-s + (−1.22 − 0.707i)11-s − 13-s + (−0.5 − 0.866i)19-s + (0.499 − 0.866i)25-s + (−0.5 + 0.866i)31-s + (1.22 + 0.707i)35-s + (−0.5 − 0.866i)37-s − 1.41i·41-s + 43-s + (−1.22 + 0.707i)47-s + (−0.499 + 0.866i)49-s + 2·55-s + (1.22 − 0.707i)65-s + ⋯

Functional equation

Λ(s)=(1008s/2 ΓC(s) L(s)=((−0.851+0.524i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1008s/2 ΓC(s) L(s)=((−0.851+0.524i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10081008    =    24⋅32⋅72^{4} \cdot 3^{2} \cdot 7
Sign: −0.851+0.524i-0.851 + 0.524i
Analytic conductor: 0.5030570.503057
Root analytic conductor: 0.7092650.709265
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1008(305,⋅)\chi_{1008} (305, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1008, ( :0), −0.851+0.524i)(2,\ 1008,\ (\ :0),\ -0.851 + 0.524i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.20107166550.2010716655
L(12)L(\frac12) ≈\approx 0.20107166550.2010716655
L(1)L(1) 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
7 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good5 1+(1.22−0.707i)T+(0.5−0.866i)T2 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}
11 1+(1.22+0.707i)T+(0.5+0.866i)T2 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2}
13 1+T+T2 1 + T + T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
37 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
41 1+1.41iT−T2 1 + 1.41iT - T^{2}
43 1−T+T2 1 - T + T^{2}
47 1+(1.22−0.707i)T+(0.5−0.866i)T2 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}
53 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
67 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
71 1+1.41iT−T2 1 + 1.41iT - T^{2}
73 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
79 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
83 1−1.41iT−T2 1 - 1.41iT - T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1+T2 1 + 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

−10.03503827289418994736538453508, −8.931203825246533758155290959458, −7.924015944353601385400201802547, −7.33139677408003243034124460075, −6.75326508655141925745908134009, −5.43690806510760131763004472120, −4.38463582948927529282666238910, −3.44798733023799191231462458514, −2.62804268978152513700220461899, −0.17505609406316021203943626829, 2.16399280714451659372404236865, 3.30163356021236317721278618921, 4.51154466921230235228734436905, 5.13567215735783345855231034502, 6.21996387356881684044387915527, 7.45994507190752733238767010073, 7.950646564031574696113835090835, 8.748819019362095814093471854051, 9.699129369716018495139620477280, 10.35669186959104996696296019564

Graph of the ZZ-function along the critical line