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Properties

Label 2-21e2-147.101-c1-0-17
Degree 22
Conductor 441441
Sign 0.239+0.971i0.239 + 0.971i
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  + (1.50 + 0.588i)2-s + (0.437 + 0.406i)4-s + (−3.18 − 2.16i)5-s + (2.37 − 1.17i)7-s + (−0.980 − 2.03i)8-s + (−3.49 − 5.12i)10-s + (−0.655 − 4.34i)11-s + (−3.43 + 2.74i)13-s + (4.24 − 0.363i)14-s + (−0.361 − 4.82i)16-s + (3.21 + 0.992i)17-s + (−0.776 + 0.448i)19-s + (−0.511 − 2.24i)20-s + (1.57 − 6.90i)22-s + (−0.732 − 2.37i)23-s + ⋯
L(s)  = 1  + (1.06 + 0.416i)2-s + (0.218 + 0.203i)4-s + (−1.42 − 0.969i)5-s + (0.896 − 0.443i)7-s + (−0.346 − 0.720i)8-s + (−1.10 − 1.62i)10-s + (−0.197 − 1.31i)11-s + (−0.953 + 0.760i)13-s + (1.13 − 0.0972i)14-s + (−0.0903 − 1.20i)16-s + (0.780 + 0.240i)17-s + (−0.178 + 0.102i)19-s + (−0.114 − 0.501i)20-s + (0.336 − 1.47i)22-s + (−0.152 − 0.494i)23-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((0.239+0.971i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1/2) L(s)=((0.239+0.971i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: 0.239+0.971i0.239 + 0.971i
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), 0.239+0.971i)(2,\ 441,\ (\ :1/2),\ 0.239 + 0.971i)

Particular Values

L(1)L(1) ≈\approx 1.29543−1.01522i1.29543 - 1.01522i
L(12)L(\frac12) ≈\approx 1.29543−1.01522i1.29543 - 1.01522i
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+(−2.37+1.17i)T 1 + (-2.37 + 1.17i)T
good2 1+(−1.50−0.588i)T+(1.46+1.36i)T2 1 + (-1.50 - 0.588i)T + (1.46 + 1.36i)T^{2}
5 1+(3.18+2.16i)T+(1.82+4.65i)T2 1 + (3.18 + 2.16i)T + (1.82 + 4.65i)T^{2}
11 1+(0.655+4.34i)T+(−10.5+3.24i)T2 1 + (0.655 + 4.34i)T + (-10.5 + 3.24i)T^{2}
13 1+(3.43−2.74i)T+(2.89−12.6i)T2 1 + (3.43 - 2.74i)T + (2.89 - 12.6i)T^{2}
17 1+(−3.21−0.992i)T+(14.0+9.57i)T2 1 + (-3.21 - 0.992i)T + (14.0 + 9.57i)T^{2}
19 1+(0.776−0.448i)T+(9.5−16.4i)T2 1 + (0.776 - 0.448i)T + (9.5 - 16.4i)T^{2}
23 1+(0.732+2.37i)T+(−19.0+12.9i)T2 1 + (0.732 + 2.37i)T + (-19.0 + 12.9i)T^{2}
29 1+(−8.30+1.89i)T+(26.1−12.5i)T2 1 + (-8.30 + 1.89i)T + (26.1 - 12.5i)T^{2}
31 1+(−4.67−2.69i)T+(15.5+26.8i)T2 1 + (-4.67 - 2.69i)T + (15.5 + 26.8i)T^{2}
37 1+(−5.45+5.06i)T+(2.76−36.8i)T2 1 + (-5.45 + 5.06i)T + (2.76 - 36.8i)T^{2}
41 1+(5.36−2.58i)T+(25.5−32.0i)T2 1 + (5.36 - 2.58i)T + (25.5 - 32.0i)T^{2}
43 1+(2.54+1.22i)T+(26.8+33.6i)T2 1 + (2.54 + 1.22i)T + (26.8 + 33.6i)T^{2}
47 1+(−1.55+3.96i)T+(−34.4−31.9i)T2 1 + (-1.55 + 3.96i)T + (-34.4 - 31.9i)T^{2}
53 1+(8.87−9.56i)T+(−3.96−52.8i)T2 1 + (8.87 - 9.56i)T + (-3.96 - 52.8i)T^{2}
59 1+(6.88−4.69i)T+(21.5−54.9i)T2 1 + (6.88 - 4.69i)T + (21.5 - 54.9i)T^{2}
61 1+(−2.93−3.16i)T+(−4.55+60.8i)T2 1 + (-2.93 - 3.16i)T + (-4.55 + 60.8i)T^{2}
67 1+(1.42−2.47i)T+(−33.5−58.0i)T2 1 + (1.42 - 2.47i)T + (-33.5 - 58.0i)T^{2}
71 1+(−7.30−1.66i)T+(63.9+30.8i)T2 1 + (-7.30 - 1.66i)T + (63.9 + 30.8i)T^{2}
73 1+(−3.88+1.52i)T+(53.5−49.6i)T2 1 + (-3.88 + 1.52i)T + (53.5 - 49.6i)T^{2}
79 1+(1.80+3.13i)T+(−39.5+68.4i)T2 1 + (1.80 + 3.13i)T + (-39.5 + 68.4i)T^{2}
83 1+(−8.34+10.4i)T+(−18.4−80.9i)T2 1 + (-8.34 + 10.4i)T + (-18.4 - 80.9i)T^{2}
89 1+(3.95+0.596i)T+(85.0+26.2i)T2 1 + (3.95 + 0.596i)T + (85.0 + 26.2i)T^{2}
97 1+18.1iT−97T2 1 + 18.1iT - 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

−11.32592880539812154930711562713, −10.13761240151840164347137895736, −8.769959909355701905043037376855, −8.109557190535243890210151642694, −7.23789840479000193314210045599, −5.97366651755077988109441585885, −4.75030235643802785458452282850, −4.45779719628557913354219840472, −3.30614709663208077857241777360, −0.77346800173218867094660964790, 2.44731730642310555427722273029, 3.32755174798230717932394957682, 4.56495107240327441967560016302, 5.05752568705339224910185911057, 6.63394743473228841639506698682, 7.85447699844047461381232011919, 8.073911635499423030574193019948, 9.797577612802556039967369420796, 10.76423283944870666013118001815, 11.66807583417981968786202190153

Graph of the ZZ-function along the critical line