↑

Properties

Label 2-882-147.104-c1-0-4
Degree 22
Conductor 882882
Sign −0.0734−0.997i-0.0734 - 0.997i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.781 + 0.623i)2-s + (0.222 + 0.974i)4-s + (1.35 + 0.651i)5-s + (1.05 + 2.42i)7-s + (−0.433 + 0.900i)8-s + (0.651 + 1.35i)10-s + (1.68 + 1.34i)11-s + (−2.26 − 1.80i)13-s + (−0.691 + 2.55i)14-s + (−0.900 + 0.433i)16-s + (−0.0455 + 0.199i)17-s + 5.39i·19-s + (−0.334 + 1.46i)20-s + (0.480 + 2.10i)22-s + (−1.31 + 0.300i)23-s + ⋯
L(s)  = 1  + (0.552 + 0.440i)2-s + (0.111 + 0.487i)4-s + (0.604 + 0.291i)5-s + (0.397 + 0.917i)7-s + (−0.153 + 0.318i)8-s + (0.206 + 0.427i)10-s + (0.508 + 0.405i)11-s + (−0.628 − 0.501i)13-s + (−0.184 + 0.682i)14-s + (−0.225 + 0.108i)16-s + (−0.0110 + 0.0483i)17-s + 1.23i·19-s + (−0.0747 + 0.327i)20-s + (0.102 + 0.448i)22-s + (−0.274 + 0.0626i)23-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=((−0.0734−0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0734 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+1/2) L(s)=((−0.0734−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0734 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: −0.0734−0.997i-0.0734 - 0.997i
Analytic conductor: 7.042807.04280
Root analytic conductor: 2.653822.65382
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ882(251,⋅)\chi_{882} (251, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 882, ( :1/2), −0.0734−0.997i)(2,\ 882,\ (\ :1/2),\ -0.0734 - 0.997i)

Particular Values

L(1)L(1) ≈\approx 1.61348+1.73671i1.61348 + 1.73671i
L(12)L(\frac12) ≈\approx 1.61348+1.73671i1.61348 + 1.73671i
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+(−0.781−0.623i)T 1 + (-0.781 - 0.623i)T
3 1 1
7 1+(−1.05−2.42i)T 1 + (-1.05 - 2.42i)T
good5 1+(−1.35−0.651i)T+(3.11+3.90i)T2 1 + (-1.35 - 0.651i)T + (3.11 + 3.90i)T^{2}
11 1+(−1.68−1.34i)T+(2.44+10.7i)T2 1 + (-1.68 - 1.34i)T + (2.44 + 10.7i)T^{2}
13 1+(2.26+1.80i)T+(2.89+12.6i)T2 1 + (2.26 + 1.80i)T + (2.89 + 12.6i)T^{2}
17 1+(0.0455−0.199i)T+(−15.3−7.37i)T2 1 + (0.0455 - 0.199i)T + (-15.3 - 7.37i)T^{2}
19 1−5.39iT−19T2 1 - 5.39iT - 19T^{2}
23 1+(1.31−0.300i)T+(20.7−9.97i)T2 1 + (1.31 - 0.300i)T + (20.7 - 9.97i)T^{2}
29 1+(−9.29−2.12i)T+(26.1+12.5i)T2 1 + (-9.29 - 2.12i)T + (26.1 + 12.5i)T^{2}
31 1+7.31iT−31T2 1 + 7.31iT - 31T^{2}
37 1+(0.0944−0.413i)T+(−33.3−16.0i)T2 1 + (0.0944 - 0.413i)T + (-33.3 - 16.0i)T^{2}
41 1+(−1.15−0.556i)T+(25.5+32.0i)T2 1 + (-1.15 - 0.556i)T + (25.5 + 32.0i)T^{2}
43 1+(8.01−3.86i)T+(26.8−33.6i)T2 1 + (8.01 - 3.86i)T + (26.8 - 33.6i)T^{2}
47 1+(2.87−3.60i)T+(−10.4−45.8i)T2 1 + (2.87 - 3.60i)T + (-10.4 - 45.8i)T^{2}
53 1+(−7.68+1.75i)T+(47.7−22.9i)T2 1 + (-7.68 + 1.75i)T + (47.7 - 22.9i)T^{2}
59 1+(−5.96+2.87i)T+(36.7−46.1i)T2 1 + (-5.96 + 2.87i)T + (36.7 - 46.1i)T^{2}
61 1+(1.14+0.260i)T+(54.9+26.4i)T2 1 + (1.14 + 0.260i)T + (54.9 + 26.4i)T^{2}
67 1+1.42T+67T2 1 + 1.42T + 67T^{2}
71 1+(−10.7+2.44i)T+(63.9−30.8i)T2 1 + (-10.7 + 2.44i)T + (63.9 - 30.8i)T^{2}
73 1+(5.67−4.52i)T+(16.2−71.1i)T2 1 + (5.67 - 4.52i)T + (16.2 - 71.1i)T^{2}
79 1−10.0T+79T2 1 - 10.0T + 79T^{2}
83 1+(−4.55−5.71i)T+(−18.4+80.9i)T2 1 + (-4.55 - 5.71i)T + (-18.4 + 80.9i)T^{2}
89 1+(4.14+5.20i)T+(−19.8+86.7i)T2 1 + (4.14 + 5.20i)T + (-19.8 + 86.7i)T^{2}
97 1+18.0iT−97T2 1 + 18.0iT - 97T^{2}
show more
show less
   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

−10.15133270455694844449404688161, −9.645664933907851826663659816800, −8.446513932984298222653708543257, −7.87184248604049994525723118997, −6.68596567442632579470243243888, −5.99226274582739406956770550092, −5.21501625491200967957606852342, −4.22347909277725910323043410744, −2.90010114637551658941791329348, −1.90764359612071326234978132699, 1.01213358866575292616218436375, 2.25552841031723085806084641032, 3.54991839723126251980207537547, 4.60490321583488173640370979550, 5.22881208967227920596421747160, 6.50501020950965967066550473230, 7.08209943712051525570404461267, 8.364160455754511380042077031030, 9.234623480360755874903247074997, 10.07078455669481799256280873684

Graph of the ZZ-function along the critical line