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Properties

Label 2-2e10-8.5-c1-0-6
Degree 22
Conductor 10241024
Sign −1-1
Analytic cond. 8.176688.17668
Root an. cond. 2.859482.85948
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  + 2.61i·3-s + 3.41i·5-s + 1.53·7-s − 3.82·9-s + 4.77i·11-s − 0.585i·13-s − 8.92·15-s − 2.82·17-s − 0.448i·19-s + 4i·21-s + 5.86·23-s − 6.65·25-s − 2.16i·27-s − 4.58i·29-s + 7.39·31-s + ⋯
L(s)  = 1  + 1.50i·3-s + 1.52i·5-s + 0.578·7-s − 1.27·9-s + 1.44i·11-s − 0.162i·13-s − 2.30·15-s − 0.685·17-s − 0.102i·19-s + 0.872i·21-s + 1.22·23-s − 1.33·25-s − 0.416i·27-s − 0.851i·29-s + 1.32·31-s + ⋯

Functional equation

Λ(s)=(1024s/2 ΓC(s) L(s)=(− Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1024s/2 ΓC(s+1/2) L(s)=(− Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10241024    =    2102^{10}
Sign: −1-1
Analytic conductor: 8.176688.17668
Root analytic conductor: 2.859482.85948
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1024(513,⋅)\chi_{1024} (513, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1024, ( :1/2), −1)(2,\ 1024,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) ≈\approx 1.5746701711.574670171
L(12)L(\frac12) ≈\approx 1.5746701711.574670171
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
good3 1−2.61iT−3T2 1 - 2.61iT - 3T^{2}
5 1−3.41iT−5T2 1 - 3.41iT - 5T^{2}
7 1−1.53T+7T2 1 - 1.53T + 7T^{2}
11 1−4.77iT−11T2 1 - 4.77iT - 11T^{2}
13 1+0.585iT−13T2 1 + 0.585iT - 13T^{2}
17 1+2.82T+17T2 1 + 2.82T + 17T^{2}
19 1+0.448iT−19T2 1 + 0.448iT - 19T^{2}
23 1−5.86T+23T2 1 - 5.86T + 23T^{2}
29 1+4.58iT−29T2 1 + 4.58iT - 29T^{2}
31 1−7.39T+31T2 1 - 7.39T + 31T^{2}
37 1+5.07iT−37T2 1 + 5.07iT - 37T^{2}
41 1−4T+41T2 1 - 4T + 41T^{2}
43 1−2.61iT−43T2 1 - 2.61iT - 43T^{2}
47 1−7.39T+47T2 1 - 7.39T + 47T^{2}
53 1−7.41iT−53T2 1 - 7.41iT - 53T^{2}
59 1+2.61iT−59T2 1 + 2.61iT - 59T^{2}
61 1+13.0iT−61T2 1 + 13.0iT - 61T^{2}
67 1+10.0iT−67T2 1 + 10.0iT - 67T^{2}
71 1+11.9T+71T2 1 + 11.9T + 71T^{2}
73 1+10.4T+73T2 1 + 10.4T + 73T^{2}
79 1−6.12T+79T2 1 - 6.12T + 79T^{2}
83 1−3.50iT−83T2 1 - 3.50iT - 83T^{2}
89 1−0.828T+89T2 1 - 0.828T + 89T^{2}
97 1−10.8T+97T2 1 - 10.8T + 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

−10.38335039989596243467772507723, −9.718874575587741754632700504515, −9.021153329597115761346955158949, −7.77936374225031226099961616133, −7.02936150945788513967802979307, −6.08541524716117397683998773820, −4.82565910318275062351544191626, −4.32885288161913632559807493785, −3.19595605589059963400216967706, −2.28332866909072352122555587152, 0.77605242185660220359820470352, 1.45758768056496823398091512716, 2.82400656246198114403080816276, 4.39054769868073767882556639139, 5.31645085544655278637911253053, 6.13142259197141154307685108635, 7.07074992061621501700243996567, 8.002747256233435195310877575609, 8.663746969624410865309688894493, 8.965492516967318121868438038053

Graph of the ZZ-function along the critical line