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Properties

Label 2-21e2-63.58-c1-0-11
Degree 22
Conductor 441441
Sign 0.967−0.253i0.967 - 0.253i
Analytic cond. 3.521403.52140
Root an. cond. 1.876541.87654
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  − 0.239·2-s + (−1.09 + 1.34i)3-s − 1.94·4-s + (−0.590 − 1.02i)5-s + (0.260 − 0.321i)6-s + 0.942·8-s + (−0.619 − 2.93i)9-s + (0.141 + 0.244i)10-s + (1.85 − 3.20i)11-s + (2.11 − 2.61i)12-s + (−0.5 + 0.866i)13-s + (2.02 + 0.321i)15-s + 3.66·16-s + (3.47 + 6.01i)17-s + (0.148 + 0.701i)18-s + (−0.971 + 1.68i)19-s + ⋯
L(s)  = 1  − 0.169·2-s + (−0.629 + 0.776i)3-s − 0.971·4-s + (−0.264 − 0.457i)5-s + (0.106 − 0.131i)6-s + 0.333·8-s + (−0.206 − 0.978i)9-s + (0.0446 + 0.0774i)10-s + (0.558 − 0.967i)11-s + (0.611 − 0.754i)12-s + (−0.138 + 0.240i)13-s + (0.522 + 0.0830i)15-s + 0.915·16-s + (0.841 + 1.45i)17-s + (0.0349 + 0.165i)18-s + (−0.222 + 0.385i)19-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((0.967−0.253i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1/2) L(s)=((0.967−0.253i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: 0.967−0.253i0.967 - 0.253i
Analytic conductor: 3.521403.52140
Root analytic conductor: 1.876541.87654
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ441(373,⋅)\chi_{441} (373, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :1/2), 0.967−0.253i)(2,\ 441,\ (\ :1/2),\ 0.967 - 0.253i)

Particular Values

L(1)L(1) ≈\approx 0.778533+0.100120i0.778533 + 0.100120i
L(12)L(\frac12) ≈\approx 0.778533+0.100120i0.778533 + 0.100120i
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)
bad3 1+(1.09−1.34i)T 1 + (1.09 - 1.34i)T
7 1 1
good2 1+0.239T+2T2 1 + 0.239T + 2T^{2}
5 1+(0.590+1.02i)T+(−2.5+4.33i)T2 1 + (0.590 + 1.02i)T + (-2.5 + 4.33i)T^{2}
11 1+(−1.85+3.20i)T+(−5.5−9.52i)T2 1 + (-1.85 + 3.20i)T + (-5.5 - 9.52i)T^{2}
13 1+(0.5−0.866i)T+(−6.5−11.2i)T2 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2}
17 1+(−3.47−6.01i)T+(−8.5+14.7i)T2 1 + (-3.47 - 6.01i)T + (-8.5 + 14.7i)T^{2}
19 1+(0.971−1.68i)T+(−9.5−16.4i)T2 1 + (0.971 - 1.68i)T + (-9.5 - 16.4i)T^{2}
23 1+(−2.80−4.85i)T+(−11.5+19.9i)T2 1 + (-2.80 - 4.85i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.119+0.207i)T+(−14.5+25.1i)T2 1 + (0.119 + 0.207i)T + (-14.5 + 25.1i)T^{2}
31 1−1.66T+31T2 1 - 1.66T + 31T^{2}
37 1+(−4.77+8.26i)T+(−18.5−32.0i)T2 1 + (-4.77 + 8.26i)T + (-18.5 - 32.0i)T^{2}
41 1+(−5.09+8.81i)T+(−20.5−35.5i)T2 1 + (-5.09 + 8.81i)T + (-20.5 - 35.5i)T^{2}
43 1+(1.11+1.92i)T+(−21.5+37.2i)T2 1 + (1.11 + 1.92i)T + (-21.5 + 37.2i)T^{2}
47 1−5.82T+47T2 1 - 5.82T + 47T^{2}
53 1+(−5.80−10.0i)T+(−26.5+45.8i)T2 1 + (-5.80 - 10.0i)T + (-26.5 + 45.8i)T^{2}
59 1−2.60T+59T2 1 - 2.60T + 59T^{2}
61 1+7.60T+61T2 1 + 7.60T + 61T^{2}
67 1−3.50T+67T2 1 - 3.50T + 67T^{2}
71 1−8.60T+71T2 1 - 8.60T + 71T^{2}
73 1+(7.57+13.1i)T+(−36.5+63.2i)T2 1 + (7.57 + 13.1i)T + (-36.5 + 63.2i)T^{2}
79 1−7.37T+79T2 1 - 7.37T + 79T^{2}
83 1+(−3.47−6.01i)T+(−41.5+71.8i)T2 1 + (-3.47 - 6.01i)T + (-41.5 + 71.8i)T^{2}
89 1+(1.37−2.37i)T+(−44.5−77.0i)T2 1 + (1.37 - 2.37i)T + (-44.5 - 77.0i)T^{2}
97 1+(3.58+6.20i)T+(−48.5+84.0i)T2 1 + (3.58 + 6.20i)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

−10.94266160791660416265082022273, −10.30586044569088187745141121607, −9.221268573517451807790188355853, −8.769832325721090714390667979125, −7.71834117838168719293670627051, −6.08691504350663686541410982176, −5.43017540144501146058442154720, −4.21143269761757994171133612556, −3.63849217342501234715875153699, −0.891407145659966308056892614784, 0.948980920962569836788598051200, 2.83732570488504292474643333126, 4.50142330143137551517289606004, 5.23659024610299446090665615220, 6.60214917456059950221503665626, 7.35019497075926602000642090743, 8.206437095890359720485972128864, 9.375297502175271972132626008563, 10.13068542863295008524292564106, 11.19588906799989752327387152468

Graph of the ZZ-function along the critical line