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Properties

Label 2-882-63.25-c1-0-3
Degree 22
Conductor 882882
Sign −0.810−0.585i-0.810 - 0.585i
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  − 2-s + (0.448 + 1.67i)3-s + 4-s + (0.258 − 0.448i)5-s + (−0.448 − 1.67i)6-s − 8-s + (−2.59 + 1.50i)9-s + (−0.258 + 0.448i)10-s + (0.732 + 1.26i)11-s + (0.448 + 1.67i)12-s + (−1.22 − 2.12i)13-s + (0.866 + 0.232i)15-s + 16-s + (−1.74 + 3.01i)17-s + (2.59 − 1.50i)18-s + (0.258 + 0.448i)19-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.258 + 0.965i)3-s + 0.5·4-s + (0.115 − 0.200i)5-s + (−0.183 − 0.683i)6-s − 0.353·8-s + (−0.866 + 0.5i)9-s + (−0.0818 + 0.141i)10-s + (0.220 + 0.382i)11-s + (0.129 + 0.482i)12-s + (−0.339 − 0.588i)13-s + (0.223 + 0.0599i)15-s + 0.250·16-s + (−0.422 + 0.731i)17-s + (0.612 − 0.353i)18-s + (0.0593 + 0.102i)19-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=((−0.810−0.585i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+1/2) L(s)=((−0.810−0.585i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: −0.810−0.585i-0.810 - 0.585i
Analytic conductor: 7.042807.04280
Root analytic conductor: 2.653822.65382
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ882(655,⋅)\chi_{882} (655, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 882, ( :1/2), −0.810−0.585i)(2,\ 882,\ (\ :1/2),\ -0.810 - 0.585i)

Particular Values

L(1)L(1) ≈\approx 0.251271+0.777053i0.251271 + 0.777053i
L(12)L(\frac12) ≈\approx 0.251271+0.777053i0.251271 + 0.777053i
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+T 1 + T
3 1+(−0.448−1.67i)T 1 + (-0.448 - 1.67i)T
7 1 1
good5 1+(−0.258+0.448i)T+(−2.5−4.33i)T2 1 + (-0.258 + 0.448i)T + (-2.5 - 4.33i)T^{2}
11 1+(−0.732−1.26i)T+(−5.5+9.52i)T2 1 + (-0.732 - 1.26i)T + (-5.5 + 9.52i)T^{2}
13 1+(1.22+2.12i)T+(−6.5+11.2i)T2 1 + (1.22 + 2.12i)T + (-6.5 + 11.2i)T^{2}
17 1+(1.74−3.01i)T+(−8.5−14.7i)T2 1 + (1.74 - 3.01i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.258−0.448i)T+(−9.5+16.4i)T2 1 + (-0.258 - 0.448i)T + (-9.5 + 16.4i)T^{2}
23 1+(3.96−6.86i)T+(−11.5−19.9i)T2 1 + (3.96 - 6.86i)T + (-11.5 - 19.9i)T^{2}
29 1+(1.36−2.36i)T+(−14.5−25.1i)T2 1 + (1.36 - 2.36i)T + (-14.5 - 25.1i)T^{2}
31 1+7.34T+31T2 1 + 7.34T + 31T^{2}
37 1+(−4−6.92i)T+(−18.5+32.0i)T2 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2}
41 1+(−2.82−4.89i)T+(−20.5+35.5i)T2 1 + (-2.82 - 4.89i)T + (-20.5 + 35.5i)T^{2}
43 1+(−6.09+10.5i)T+(−21.5−37.2i)T2 1 + (-6.09 + 10.5i)T + (-21.5 - 37.2i)T^{2}
47 1−4.62T+47T2 1 - 4.62T + 47T^{2}
53 1+(3.36−5.83i)T+(−26.5−45.8i)T2 1 + (3.36 - 5.83i)T + (-26.5 - 45.8i)T^{2}
59 1+14.7T+59T2 1 + 14.7T + 59T^{2}
61 1+4.38T+61T2 1 + 4.38T + 61T^{2}
67 1+3.80T+67T2 1 + 3.80T + 67T^{2}
71 1+0.803T+71T2 1 + 0.803T + 71T^{2}
73 1+(−2.31+4.00i)T+(−36.5−63.2i)T2 1 + (-2.31 + 4.00i)T + (-36.5 - 63.2i)T^{2}
79 1−14.1T+79T2 1 - 14.1T + 79T^{2}
83 1+(4.94−8.57i)T+(−41.5−71.8i)T2 1 + (4.94 - 8.57i)T + (-41.5 - 71.8i)T^{2}
89 1+(8.05+13.9i)T+(−44.5+77.0i)T2 1 + (8.05 + 13.9i)T + (-44.5 + 77.0i)T^{2}
97 1+(0.517−0.896i)T+(−48.5−84.0i)T2 1 + (0.517 - 0.896i)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

−10.43646891986034301497745765608, −9.373367344682364316645992086725, −9.182938975454976298962256430038, −8.032637924608084753047677428695, −7.39961425398731947774732745916, −6.04210811784696792828134057470, −5.25279053466382495883594220176, −4.09051029625363667770854670515, −3.09262777402254045510598126952, −1.74046208870896397572591369528, 0.45682979462238907134217828964, 2.01268948403321958875719025800, 2.82291491530787779788584744884, 4.30424711503623088028217870070, 5.84797457069924971107875099266, 6.53912735587648309434833705249, 7.33440240416787074848018807083, 8.065083631831219895704612513250, 9.003747607866980150421314205214, 9.483632590333182667649512398054

Graph of the ZZ-function along the critical line