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Properties

Label 2-21e2-147.101-c1-0-10
Degree 22
Conductor 441441
Sign 0.998+0.0614i0.998 + 0.0614i
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  + (1.02 + 0.401i)2-s + (−0.580 − 0.538i)4-s + (1.00 + 0.687i)5-s + (2.62 − 0.364i)7-s + (−1.33 − 2.76i)8-s + (0.755 + 1.10i)10-s + (0.0890 + 0.590i)11-s + (2.95 − 2.35i)13-s + (2.82 + 0.679i)14-s + (−0.133 − 1.78i)16-s + (2.61 + 0.808i)17-s + (0.127 − 0.0734i)19-s + (−0.214 − 0.941i)20-s + (−0.146 + 0.640i)22-s + (2.07 + 6.71i)23-s + ⋯
L(s)  = 1  + (0.723 + 0.283i)2-s + (−0.290 − 0.269i)4-s + (0.450 + 0.307i)5-s + (0.990 − 0.137i)7-s + (−0.470 − 0.977i)8-s + (0.238 + 0.350i)10-s + (0.0268 + 0.178i)11-s + (0.819 − 0.653i)13-s + (0.755 + 0.181i)14-s + (−0.0334 − 0.446i)16-s + (0.635 + 0.195i)17-s + (0.0292 − 0.0168i)19-s + (−0.0480 − 0.210i)20-s + (−0.0311 + 0.136i)22-s + (0.431 + 1.39i)23-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((0.998+0.0614i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0614i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1/2) L(s)=((0.998+0.0614i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0614i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: 0.998+0.0614i0.998 + 0.0614i
Analytic conductor: 3.521403.52140
Root analytic conductor: 1.876541.87654
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ441(395,⋅)\chi_{441} (395, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :1/2), 0.998+0.0614i)(2,\ 441,\ (\ :1/2),\ 0.998 + 0.0614i)

Particular Values

L(1)L(1) ≈\approx 2.12929−0.0654450i2.12929 - 0.0654450i
L(12)L(\frac12) ≈\approx 2.12929−0.0654450i2.12929 - 0.0654450i
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
7 1+(−2.62+0.364i)T 1 + (-2.62 + 0.364i)T
good2 1+(−1.02−0.401i)T+(1.46+1.36i)T2 1 + (-1.02 - 0.401i)T + (1.46 + 1.36i)T^{2}
5 1+(−1.00−0.687i)T+(1.82+4.65i)T2 1 + (-1.00 - 0.687i)T + (1.82 + 4.65i)T^{2}
11 1+(−0.0890−0.590i)T+(−10.5+3.24i)T2 1 + (-0.0890 - 0.590i)T + (-10.5 + 3.24i)T^{2}
13 1+(−2.95+2.35i)T+(2.89−12.6i)T2 1 + (-2.95 + 2.35i)T + (2.89 - 12.6i)T^{2}
17 1+(−2.61−0.808i)T+(14.0+9.57i)T2 1 + (-2.61 - 0.808i)T + (14.0 + 9.57i)T^{2}
19 1+(−0.127+0.0734i)T+(9.5−16.4i)T2 1 + (-0.127 + 0.0734i)T + (9.5 - 16.4i)T^{2}
23 1+(−2.07−6.71i)T+(−19.0+12.9i)T2 1 + (-2.07 - 6.71i)T + (-19.0 + 12.9i)T^{2}
29 1+(−2.75+0.628i)T+(26.1−12.5i)T2 1 + (-2.75 + 0.628i)T + (26.1 - 12.5i)T^{2}
31 1+(4.54+2.62i)T+(15.5+26.8i)T2 1 + (4.54 + 2.62i)T + (15.5 + 26.8i)T^{2}
37 1+(0.323−0.300i)T+(2.76−36.8i)T2 1 + (0.323 - 0.300i)T + (2.76 - 36.8i)T^{2}
41 1+(4.98−2.39i)T+(25.5−32.0i)T2 1 + (4.98 - 2.39i)T + (25.5 - 32.0i)T^{2}
43 1+(−1.07−0.517i)T+(26.8+33.6i)T2 1 + (-1.07 - 0.517i)T + (26.8 + 33.6i)T^{2}
47 1+(−1.29+3.30i)T+(−34.4−31.9i)T2 1 + (-1.29 + 3.30i)T + (-34.4 - 31.9i)T^{2}
53 1+(8.63−9.30i)T+(−3.96−52.8i)T2 1 + (8.63 - 9.30i)T + (-3.96 - 52.8i)T^{2}
59 1+(1.25−0.858i)T+(21.5−54.9i)T2 1 + (1.25 - 0.858i)T + (21.5 - 54.9i)T^{2}
61 1+(9.59+10.3i)T+(−4.55+60.8i)T2 1 + (9.59 + 10.3i)T + (-4.55 + 60.8i)T^{2}
67 1+(1.69−2.93i)T+(−33.5−58.0i)T2 1 + (1.69 - 2.93i)T + (-33.5 - 58.0i)T^{2}
71 1+(6.84+1.56i)T+(63.9+30.8i)T2 1 + (6.84 + 1.56i)T + (63.9 + 30.8i)T^{2}
73 1+(14.5−5.69i)T+(53.5−49.6i)T2 1 + (14.5 - 5.69i)T + (53.5 - 49.6i)T^{2}
79 1+(−5.62−9.74i)T+(−39.5+68.4i)T2 1 + (-5.62 - 9.74i)T + (-39.5 + 68.4i)T^{2}
83 1+(4.09−5.14i)T+(−18.4−80.9i)T2 1 + (4.09 - 5.14i)T + (-18.4 - 80.9i)T^{2}
89 1+(3.27+0.493i)T+(85.0+26.2i)T2 1 + (3.27 + 0.493i)T + (85.0 + 26.2i)T^{2}
97 1−12.1iT−97T2 1 - 12.1iT - 97T^{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

−11.07926314507972821180382074210, −10.26109339346921102931162649797, −9.403452246441071383468549934053, −8.302313215447557977644546195603, −7.31626400082150071699502593648, −6.07730122234566828517903319570, −5.45332617919027766933063688560, −4.42103857433293739910532267498, −3.28213319612047592984934893276, −1.42012989970481559285994747240, 1.69835457465879813039446547194, 3.17175067727806713209055152401, 4.39384720924330396785213641291, 5.15853817449908670779532976428, 6.11360234420267403905945012349, 7.53973749545705054587993514855, 8.632281898566481898569022304130, 9.034001956269373956723837921271, 10.46679398187604612527311370522, 11.35120484489233600227611763501

Graph of the ZZ-function along the critical line