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Properties

Label 2-882-63.20-c1-0-29
Degree 22
Conductor 882882
Sign 0.147−0.989i0.147 - 0.989i
Analytic cond. 7.042807.04280
Root an. cond. 2.653822.65382
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  + (0.866 + 0.5i)2-s + (1.71 + 0.264i)3-s + (0.499 + 0.866i)4-s + (1.77 + 3.07i)5-s + (1.35 + 1.08i)6-s + 0.999i·8-s + (2.85 + 0.906i)9-s + 3.55i·10-s + (−2.61 − 1.51i)11-s + (0.626 + 1.61i)12-s + (−0.888 + 0.513i)13-s + (2.22 + 5.73i)15-s + (−0.5 + 0.866i)16-s + 1.61·17-s + (2.02 + 2.21i)18-s − 8.22i·19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.988 + 0.152i)3-s + (0.249 + 0.433i)4-s + (0.794 + 1.37i)5-s + (0.551 + 0.442i)6-s + 0.353i·8-s + (0.953 + 0.302i)9-s + 1.12i·10-s + (−0.789 − 0.455i)11-s + (0.180 + 0.466i)12-s + (−0.246 + 0.142i)13-s + (0.574 + 1.48i)15-s + (−0.125 + 0.216i)16-s + 0.392·17-s + (0.476 + 0.522i)18-s − 1.88i·19-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=((0.147−0.989i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+1/2) L(s)=((0.147−0.989i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: 0.147−0.989i0.147 - 0.989i
Analytic conductor: 7.042807.04280
Root analytic conductor: 2.653822.65382
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ882(587,⋅)\chi_{882} (587, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 882, ( :1/2), 0.147−0.989i)(2,\ 882,\ (\ :1/2),\ 0.147 - 0.989i)

Particular Values

L(1)L(1) ≈\approx 2.56932+2.21485i2.56932 + 2.21485i
L(12)L(\frac12) ≈\approx 2.56932+2.21485i2.56932 + 2.21485i
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+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
3 1+(−1.71−0.264i)T 1 + (-1.71 - 0.264i)T
7 1 1
good5 1+(−1.77−3.07i)T+(−2.5+4.33i)T2 1 + (-1.77 - 3.07i)T + (-2.5 + 4.33i)T^{2}
11 1+(2.61+1.51i)T+(5.5+9.52i)T2 1 + (2.61 + 1.51i)T + (5.5 + 9.52i)T^{2}
13 1+(0.888−0.513i)T+(6.5−11.2i)T2 1 + (0.888 - 0.513i)T + (6.5 - 11.2i)T^{2}
17 1−1.61T+17T2 1 - 1.61T + 17T^{2}
19 1+8.22iT−19T2 1 + 8.22iT - 19T^{2}
23 1+(2.90−1.67i)T+(11.5−19.9i)T2 1 + (2.90 - 1.67i)T + (11.5 - 19.9i)T^{2}
29 1+(3.70+2.13i)T+(14.5+25.1i)T2 1 + (3.70 + 2.13i)T + (14.5 + 25.1i)T^{2}
31 1+(−5.18+2.99i)T+(15.5−26.8i)T2 1 + (-5.18 + 2.99i)T + (15.5 - 26.8i)T^{2}
37 1+5.84T+37T2 1 + 5.84T + 37T^{2}
41 1+(−0.0472−0.0817i)T+(−20.5+35.5i)T2 1 + (-0.0472 - 0.0817i)T + (-20.5 + 35.5i)T^{2}
43 1+(−3.05+5.29i)T+(−21.5−37.2i)T2 1 + (-3.05 + 5.29i)T + (-21.5 - 37.2i)T^{2}
47 1+(−2.57+4.45i)T+(−23.5−40.7i)T2 1 + (-2.57 + 4.45i)T + (-23.5 - 40.7i)T^{2}
53 1−3.18iT−53T2 1 - 3.18iT - 53T^{2}
59 1+(−4.42−7.65i)T+(−29.5+51.0i)T2 1 + (-4.42 - 7.65i)T + (-29.5 + 51.0i)T^{2}
61 1+(4.06+2.34i)T+(30.5+52.8i)T2 1 + (4.06 + 2.34i)T + (30.5 + 52.8i)T^{2}
67 1+(−0.187−0.325i)T+(−33.5+58.0i)T2 1 + (-0.187 - 0.325i)T + (-33.5 + 58.0i)T^{2}
71 1−13.9iT−71T2 1 - 13.9iT - 71T^{2}
73 1+1.31iT−73T2 1 + 1.31iT - 73T^{2}
79 1+(0.462−0.800i)T+(−39.5−68.4i)T2 1 + (0.462 - 0.800i)T + (-39.5 - 68.4i)T^{2}
83 1+(−5.43+9.40i)T+(−41.5−71.8i)T2 1 + (-5.43 + 9.40i)T + (-41.5 - 71.8i)T^{2}
89 1−4.70T+89T2 1 - 4.70T + 89T^{2}
97 1+(−13.3−7.69i)T+(48.5+84.0i)T2 1 + (-13.3 - 7.69i)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

−10.31773670558876871144786385826, −9.529283269116551960985485763904, −8.589699031168065934454568070552, −7.51606981036690370624126612465, −7.02479175206100706799205836595, −6.04317589266762235838329268686, −5.06486690205397779167489037593, −3.81526367271740658324772454686, −2.80768051523945549033807301908, −2.29647793179910922339432707762, 1.38954440153447791948924196278, 2.22096826051998805853263170167, 3.50421263962333652713603767567, 4.56959320795389734518506662794, 5.34571046014919679774916131859, 6.27151988320243374154414415192, 7.64868620487261135010927397254, 8.265114080787718883099877206086, 9.210146063877125594572872571753, 9.958178518452828446617928797956

Graph of the ZZ-function along the critical line