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Properties

Label 2-21e2-147.104-c1-0-14
Degree 22
Conductor 441441
Sign 0.989−0.147i0.989 - 0.147i
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  + (0.947 + 0.755i)2-s + (−0.118 − 0.518i)4-s + (3.15 + 1.52i)5-s + (1.53 − 2.15i)7-s + (1.33 − 2.76i)8-s + (1.84 + 3.82i)10-s + (−0.959 − 0.765i)11-s + (−4.81 − 3.84i)13-s + (3.08 − 0.883i)14-s + (2.39 − 1.15i)16-s + (−0.101 + 0.445i)17-s + 8.04i·19-s + (0.414 − 1.81i)20-s + (−0.330 − 1.44i)22-s + (−1.94 + 0.443i)23-s + ⋯
L(s)  = 1  + (0.669 + 0.534i)2-s + (−0.0591 − 0.259i)4-s + (1.41 + 0.679i)5-s + (0.579 − 0.814i)7-s + (0.470 − 0.977i)8-s + (0.582 + 1.20i)10-s + (−0.289 − 0.230i)11-s + (−1.33 − 1.06i)13-s + (0.823 − 0.236i)14-s + (0.597 − 0.287i)16-s + (−0.0246 + 0.107i)17-s + 1.84i·19-s + (0.0926 − 0.405i)20-s + (−0.0705 − 0.309i)22-s + (−0.405 + 0.0924i)23-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((0.989−0.147i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1/2) L(s)=((0.989−0.147i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: 0.989−0.147i0.989 - 0.147i
Analytic conductor: 3.521403.52140
Root analytic conductor: 1.876541.87654
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ441(251,⋅)\chi_{441} (251, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :1/2), 0.989−0.147i)(2,\ 441,\ (\ :1/2),\ 0.989 - 0.147i)

Particular Values

L(1)L(1) ≈\approx 2.37226+0.175820i2.37226 + 0.175820i
L(12)L(\frac12) ≈\approx 2.37226+0.175820i2.37226 + 0.175820i
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+(−1.53+2.15i)T 1 + (-1.53 + 2.15i)T
good2 1+(−0.947−0.755i)T+(0.445+1.94i)T2 1 + (-0.947 - 0.755i)T + (0.445 + 1.94i)T^{2}
5 1+(−3.15−1.52i)T+(3.11+3.90i)T2 1 + (-3.15 - 1.52i)T + (3.11 + 3.90i)T^{2}
11 1+(0.959+0.765i)T+(2.44+10.7i)T2 1 + (0.959 + 0.765i)T + (2.44 + 10.7i)T^{2}
13 1+(4.81+3.84i)T+(2.89+12.6i)T2 1 + (4.81 + 3.84i)T + (2.89 + 12.6i)T^{2}
17 1+(0.101−0.445i)T+(−15.3−7.37i)T2 1 + (0.101 - 0.445i)T + (-15.3 - 7.37i)T^{2}
19 1−8.04iT−19T2 1 - 8.04iT - 19T^{2}
23 1+(1.94−0.443i)T+(20.7−9.97i)T2 1 + (1.94 - 0.443i)T + (20.7 - 9.97i)T^{2}
29 1+(−6.85−1.56i)T+(26.1+12.5i)T2 1 + (-6.85 - 1.56i)T + (26.1 + 12.5i)T^{2}
31 1−6.87iT−31T2 1 - 6.87iT - 31T^{2}
37 1+(−0.226+0.994i)T+(−33.3−16.0i)T2 1 + (-0.226 + 0.994i)T + (-33.3 - 16.0i)T^{2}
41 1+(−5.35−2.57i)T+(25.5+32.0i)T2 1 + (-5.35 - 2.57i)T + (25.5 + 32.0i)T^{2}
43 1+(8.06−3.88i)T+(26.8−33.6i)T2 1 + (8.06 - 3.88i)T + (26.8 - 33.6i)T^{2}
47 1+(0.922−1.15i)T+(−10.4−45.8i)T2 1 + (0.922 - 1.15i)T + (-10.4 - 45.8i)T^{2}
53 1+(7.14−1.63i)T+(47.7−22.9i)T2 1 + (7.14 - 1.63i)T + (47.7 - 22.9i)T^{2}
59 1+(−6.83+3.29i)T+(36.7−46.1i)T2 1 + (-6.83 + 3.29i)T + (36.7 - 46.1i)T^{2}
61 1+(6.28+1.43i)T+(54.9+26.4i)T2 1 + (6.28 + 1.43i)T + (54.9 + 26.4i)T^{2}
67 1−1.41T+67T2 1 - 1.41T + 67T^{2}
71 1+(14.0−3.19i)T+(63.9−30.8i)T2 1 + (14.0 - 3.19i)T + (63.9 - 30.8i)T^{2}
73 1+(1.23−0.982i)T+(16.2−71.1i)T2 1 + (1.23 - 0.982i)T + (16.2 - 71.1i)T^{2}
79 1+6.59T+79T2 1 + 6.59T + 79T^{2}
83 1+(−3.24−4.06i)T+(−18.4+80.9i)T2 1 + (-3.24 - 4.06i)T + (-18.4 + 80.9i)T^{2}
89 1+(−8.27−10.3i)T+(−19.8+86.7i)T2 1 + (-8.27 - 10.3i)T + (-19.8 + 86.7i)T^{2}
97 1+11.9iT−97T2 1 + 11.9iT - 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

−10.76670536653207425739239725382, −10.11957756218760366632647653600, −9.871884094109929366131330266244, −8.135120476151902385573713635183, −7.20872167140962084545756761872, −6.25646490869361571989711892453, −5.50205142352619327525164866695, −4.66258284039593084798600609682, −3.13572380418150596217536068699, −1.56216165859924007129657255100, 2.05778525611358083836189314747, 2.57124060802740261341072074975, 4.73089866805674626071080375052, 4.85733581064152628251577569715, 6.09538898538093573876877157623, 7.41301404096876747374322825824, 8.664207014755422645505094793775, 9.266774901895640367758611769140, 10.17996138590828738755365109004, 11.48600321105428064938197555771

Graph of the ZZ-function along the critical line