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Properties

Label 2-882-9.4-c1-0-9
Degree 22
Conductor 882882
Sign −0.262−0.964i-0.262 - 0.964i
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  + (−0.5 − 0.866i)2-s + (0.933 + 1.45i)3-s + (−0.499 + 0.866i)4-s + (−0.230 + 0.398i)5-s + (0.796 − 1.53i)6-s + 0.999·8-s + (−1.25 + 2.72i)9-s + 0.460·10-s + (1.82 + 3.15i)11-s + (−1.73 + 0.0789i)12-s + (−0.730 + 1.26i)13-s + (−0.796 + 0.0363i)15-s + (−0.5 − 0.866i)16-s − 3.73·17-s + (2.98 − 0.273i)18-s − 4.05·19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.538 + 0.842i)3-s + (−0.249 + 0.433i)4-s + (−0.102 + 0.178i)5-s + (0.325 − 0.627i)6-s + 0.353·8-s + (−0.419 + 0.907i)9-s + 0.145·10-s + (0.549 + 0.952i)11-s + (−0.499 + 0.0227i)12-s + (−0.202 + 0.350i)13-s + (−0.205 + 0.00938i)15-s + (−0.125 − 0.216i)16-s − 0.905·17-s + (0.704 − 0.0643i)18-s − 0.930·19-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=((−0.262−0.964i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+1/2) L(s)=((−0.262−0.964i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: −0.262−0.964i-0.262 - 0.964i
Analytic conductor: 7.042807.04280
Root analytic conductor: 2.653822.65382
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ882(589,⋅)\chi_{882} (589, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 882, ( :1/2), −0.262−0.964i)(2,\ 882,\ (\ :1/2),\ -0.262 - 0.964i)

Particular Values

L(1)L(1) ≈\approx 0.670558+0.877408i0.670558 + 0.877408i
L(12)L(\frac12) ≈\approx 0.670558+0.877408i0.670558 + 0.877408i
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
3 1+(−0.933−1.45i)T 1 + (-0.933 - 1.45i)T
7 1 1
good5 1+(0.230−0.398i)T+(−2.5−4.33i)T2 1 + (0.230 - 0.398i)T + (-2.5 - 4.33i)T^{2}
11 1+(−1.82−3.15i)T+(−5.5+9.52i)T2 1 + (-1.82 - 3.15i)T + (-5.5 + 9.52i)T^{2}
13 1+(0.730−1.26i)T+(−6.5−11.2i)T2 1 + (0.730 - 1.26i)T + (-6.5 - 11.2i)T^{2}
17 1+3.73T+17T2 1 + 3.73T + 17T^{2}
19 1+4.05T+19T2 1 + 4.05T + 19T^{2}
23 1+(0.566−0.981i)T+(−11.5−19.9i)T2 1 + (0.566 - 0.981i)T + (-11.5 - 19.9i)T^{2}
29 1+(4.48+7.77i)T+(−14.5+25.1i)T2 1 + (4.48 + 7.77i)T + (-14.5 + 25.1i)T^{2}
31 1+(0.257−0.445i)T+(−15.5−26.8i)T2 1 + (0.257 - 0.445i)T + (-15.5 - 26.8i)T^{2}
37 1−9.10T+37T2 1 - 9.10T + 37T^{2}
41 1+(−0.472+0.819i)T+(−20.5−35.5i)T2 1 + (-0.472 + 0.819i)T + (-20.5 - 35.5i)T^{2}
43 1+(−4.66−8.07i)T+(−21.5+37.2i)T2 1 + (-4.66 - 8.07i)T + (-21.5 + 37.2i)T^{2}
47 1+(−1.16−2.01i)T+(−23.5+40.7i)T2 1 + (-1.16 - 2.01i)T + (-23.5 + 40.7i)T^{2}
53 1+12.4T+53T2 1 + 12.4T + 53T^{2}
59 1+(6.44−11.1i)T+(−29.5−51.0i)T2 1 + (6.44 - 11.1i)T + (-29.5 - 51.0i)T^{2}
61 1+(−6.04−10.4i)T+(−30.5+52.8i)T2 1 + (-6.04 - 10.4i)T + (-30.5 + 52.8i)T^{2}
67 1+(−1.16+2.00i)T+(−33.5−58.0i)T2 1 + (-1.16 + 2.00i)T + (-33.5 - 58.0i)T^{2}
71 1−1.67T+71T2 1 - 1.67T + 71T^{2}
73 1+13.2T+73T2 1 + 13.2T + 73T^{2}
79 1+(−2.50−4.33i)T+(−39.5+68.4i)T2 1 + (-2.50 - 4.33i)T + (-39.5 + 68.4i)T^{2}
83 1+(3.32+5.75i)T+(−41.5+71.8i)T2 1 + (3.32 + 5.75i)T + (-41.5 + 71.8i)T^{2}
89 1+2.72T+89T2 1 + 2.72T + 89T^{2}
97 1+(−5.59−9.68i)T+(−48.5+84.0i)T2 1 + (-5.59 - 9.68i)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

−10.29536545839241275359206481567, −9.420208052635147185605810747714, −9.123737236654428185013057852911, −8.024437178887636210874022040893, −7.24320818642201148974568093051, −6.02455957268061562223062968438, −4.48107074577826830496612455077, −4.20788426783118535793088224913, −2.85472356020979899481084827048, −1.90669855114734031407795588638, 0.54296962872212431414735931939, 2.02134690490939877612695741801, 3.35931554086640352305274923772, 4.57515989992684649891162188458, 5.91959236632380549449445389084, 6.52404120009143073811586823437, 7.36968797455707803002139284365, 8.320230295888336430410853280387, 8.755626878802998613516509744987, 9.517620052073149239717092062202

Graph of the ZZ-function along the critical line