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Properties

Label 2-21e2-63.4-c1-0-5
Degree 22
Conductor 441441
Sign 0.0644−0.997i0.0644 - 0.997i
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  + (−0.119 + 0.207i)2-s + (−0.578 − 1.63i)3-s + (0.971 + 1.68i)4-s − 2.59·5-s + (0.407 + 0.0753i)6-s − 0.942·8-s + (−2.33 + 1.88i)9-s + (0.309 − 0.536i)10-s + 4.18·11-s + (2.18 − 2.55i)12-s + (−1.84 + 3.18i)13-s + (1.5 + 4.23i)15-s + (−1.83 + 3.16i)16-s + (−0.855 + 1.48i)17-s + (−0.112 − 0.708i)18-s + (3.57 + 6.19i)19-s + ⋯
L(s)  = 1  + (−0.0845 + 0.146i)2-s + (−0.334 − 0.942i)3-s + (0.485 + 0.841i)4-s − 1.15·5-s + (0.166 + 0.0307i)6-s − 0.333·8-s + (−0.776 + 0.629i)9-s + (0.0979 − 0.169i)10-s + 1.26·11-s + (0.630 − 0.738i)12-s + (−0.510 + 0.884i)13-s + (0.387 + 1.09i)15-s + (−0.457 + 0.792i)16-s + (−0.207 + 0.359i)17-s + (−0.0265 − 0.166i)18-s + (0.820 + 1.42i)19-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((0.0644−0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0644 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1/2) L(s)=((0.0644−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0644 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: 0.0644−0.997i0.0644 - 0.997i
Analytic conductor: 3.521403.52140
Root analytic conductor: 1.876541.87654
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ441(67,⋅)\chi_{441} (67, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :1/2), 0.0644−0.997i)(2,\ 441,\ (\ :1/2),\ 0.0644 - 0.997i)

Particular Values

L(1)L(1) ≈\approx 0.606799+0.568857i0.606799 + 0.568857i
L(12)L(\frac12) ≈\approx 0.606799+0.568857i0.606799 + 0.568857i
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+(0.578+1.63i)T 1 + (0.578 + 1.63i)T
7 1 1
good2 1+(0.119−0.207i)T+(−1−1.73i)T2 1 + (0.119 - 0.207i)T + (-1 - 1.73i)T^{2}
5 1+2.59T+5T2 1 + 2.59T + 5T^{2}
11 1−4.18T+11T2 1 - 4.18T + 11T^{2}
13 1+(1.84−3.18i)T+(−6.5−11.2i)T2 1 + (1.84 - 3.18i)T + (-6.5 - 11.2i)T^{2}
17 1+(0.855−1.48i)T+(−8.5−14.7i)T2 1 + (0.855 - 1.48i)T + (-8.5 - 14.7i)T^{2}
19 1+(−3.57−6.19i)T+(−9.5+16.4i)T2 1 + (-3.57 - 6.19i)T + (-9.5 + 16.4i)T^{2}
23 1+5.12T+23T2 1 + 5.12T + 23T^{2}
29 1+(−1.06−1.84i)T+(−14.5+25.1i)T2 1 + (-1.06 - 1.84i)T + (-14.5 + 25.1i)T^{2}
31 1+(−3.26−5.66i)T+(−15.5+26.8i)T2 1 + (-3.26 - 5.66i)T + (-15.5 + 26.8i)T^{2}
37 1+(0.830+1.43i)T+(−18.5+32.0i)T2 1 + (0.830 + 1.43i)T + (-18.5 + 32.0i)T^{2}
41 1+(−5.10+8.84i)T+(−20.5−35.5i)T2 1 + (-5.10 + 8.84i)T + (-20.5 - 35.5i)T^{2}
43 1+(−0.830−1.43i)T+(−21.5+37.2i)T2 1 + (-0.830 - 1.43i)T + (-21.5 + 37.2i)T^{2}
47 1+(4.66−8.08i)T+(−23.5−40.7i)T2 1 + (4.66 - 8.08i)T + (-23.5 - 40.7i)T^{2}
53 1+(5.32−9.22i)T+(−26.5−45.8i)T2 1 + (5.32 - 9.22i)T + (-26.5 - 45.8i)T^{2}
59 1+(3.03+5.25i)T+(−29.5+51.0i)T2 1 + (3.03 + 5.25i)T + (-29.5 + 51.0i)T^{2}
61 1+(−3.99+6.91i)T+(−30.5−52.8i)T2 1 + (-3.99 + 6.91i)T + (-30.5 - 52.8i)T^{2}
67 1+(4.13+7.15i)T+(−33.5+58.0i)T2 1 + (4.13 + 7.15i)T + (-33.5 + 58.0i)T^{2}
71 1−6.23T+71T2 1 - 6.23T + 71T^{2}
73 1+(3.57−6.19i)T+(−36.5−63.2i)T2 1 + (3.57 - 6.19i)T + (-36.5 - 63.2i)T^{2}
79 1+(−4.91+8.51i)T+(−39.5−68.4i)T2 1 + (-4.91 + 8.51i)T + (-39.5 - 68.4i)T^{2}
83 1+(3.44+5.97i)T+(−41.5+71.8i)T2 1 + (3.44 + 5.97i)T + (-41.5 + 71.8i)T^{2}
89 1+(−2.51−4.36i)T+(−44.5+77.0i)T2 1 + (-2.51 - 4.36i)T + (-44.5 + 77.0i)T^{2}
97 1+(1.53+2.65i)T+(−48.5+84.0i)T2 1 + (1.53 + 2.65i)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

−11.70480679064857244086457987392, −10.84174506107433101473124412616, −9.274798424537211570069562400597, −8.220365144850365074693957696089, −7.66017402647434378136957482754, −6.85342791521473604959348581628, −6.07522257297866300903922257496, −4.32176914452828040478614350223, −3.37275197335692125292191043693, −1.74566544168349118372809125927, 0.56352934482931732821145294547, 2.85477615780255834684628357904, 4.08139871661966041964412482614, 5.01020561084670447522089781971, 6.11796399434232449266824807800, 7.09407373770462777620943728789, 8.275902012548167300613256031028, 9.468169834503790365724301997175, 9.934639573078498773682551843855, 11.08935336682695214667288315191

Graph of the ZZ-function along the critical line