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Properties

Label 2-21e2-63.25-c1-0-23
Degree 22
Conductor 441441
Sign −0.968−0.250i-0.968 - 0.250i
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  − 0.879·2-s + (−1.70 − 0.300i)3-s − 1.22·4-s + (0.673 − 1.16i)5-s + (1.49 + 0.264i)6-s + 2.83·8-s + (2.81 + 1.02i)9-s + (−0.592 + 1.02i)10-s + (−0.826 − 1.43i)11-s + (2.09 + 0.368i)12-s + (−1.68 − 2.91i)13-s + (−1.49 + 1.78i)15-s − 0.0418·16-s + (0.233 − 0.405i)17-s + (−2.47 − 0.902i)18-s + (−1.61 − 2.79i)19-s + ⋯
L(s)  = 1  − 0.621·2-s + (−0.984 − 0.173i)3-s − 0.613·4-s + (0.301 − 0.521i)5-s + (0.612 + 0.107i)6-s + 1.00·8-s + (0.939 + 0.342i)9-s + (−0.187 + 0.324i)10-s + (−0.249 − 0.431i)11-s + (0.604 + 0.106i)12-s + (−0.467 − 0.809i)13-s + (−0.387 + 0.461i)15-s − 0.0104·16-s + (0.0567 − 0.0982i)17-s + (−0.584 − 0.212i)18-s + (−0.370 − 0.641i)19-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((−0.968−0.250i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1/2) L(s)=((−0.968−0.250i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: −0.968−0.250i-0.968 - 0.250i
Analytic conductor: 3.521403.52140
Root analytic conductor: 1.876541.87654
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ441(214,⋅)\chi_{441} (214, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :1/2), −0.968−0.250i)(2,\ 441,\ (\ :1/2),\ -0.968 - 0.250i)

Particular Values

L(1)L(1) ≈\approx 0.0107471+0.0843349i0.0107471 + 0.0843349i
L(12)L(\frac12) ≈\approx 0.0107471+0.0843349i0.0107471 + 0.0843349i
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+(1.70+0.300i)T 1 + (1.70 + 0.300i)T
7 1 1
good2 1+0.879T+2T2 1 + 0.879T + 2T^{2}
5 1+(−0.673+1.16i)T+(−2.5−4.33i)T2 1 + (-0.673 + 1.16i)T + (-2.5 - 4.33i)T^{2}
11 1+(0.826+1.43i)T+(−5.5+9.52i)T2 1 + (0.826 + 1.43i)T + (-5.5 + 9.52i)T^{2}
13 1+(1.68+2.91i)T+(−6.5+11.2i)T2 1 + (1.68 + 2.91i)T + (-6.5 + 11.2i)T^{2}
17 1+(−0.233+0.405i)T+(−8.5−14.7i)T2 1 + (-0.233 + 0.405i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.61+2.79i)T+(−9.5+16.4i)T2 1 + (1.61 + 2.79i)T + (-9.5 + 16.4i)T^{2}
23 1+(4.47−7.74i)T+(−11.5−19.9i)T2 1 + (4.47 - 7.74i)T + (-11.5 - 19.9i)T^{2}
29 1+(3.13−5.42i)T+(−14.5−25.1i)T2 1 + (3.13 - 5.42i)T + (-14.5 - 25.1i)T^{2}
31 1+9.23T+31T2 1 + 9.23T + 31T^{2}
37 1+(4.61+7.99i)T+(−18.5+32.0i)T2 1 + (4.61 + 7.99i)T + (-18.5 + 32.0i)T^{2}
41 1+(−1.70−2.95i)T+(−20.5+35.5i)T2 1 + (-1.70 - 2.95i)T + (-20.5 + 35.5i)T^{2}
43 1+(−2.20+3.82i)T+(−21.5−37.2i)T2 1 + (-2.20 + 3.82i)T + (-21.5 - 37.2i)T^{2}
47 1+9.35T+47T2 1 + 9.35T + 47T^{2}
53 1+(−0.286+0.497i)T+(−26.5−45.8i)T2 1 + (-0.286 + 0.497i)T + (-26.5 - 45.8i)T^{2}
59 1−10.3T+59T2 1 - 10.3T + 59T^{2}
61 1+7.63T+61T2 1 + 7.63T + 61T^{2}
67 1−0.596T+67T2 1 - 0.596T + 67T^{2}
71 1+0.554T+71T2 1 + 0.554T + 71T^{2}
73 1+(−1.02+1.77i)T+(−36.5−63.2i)T2 1 + (-1.02 + 1.77i)T + (-36.5 - 63.2i)T^{2}
79 1+2.40T+79T2 1 + 2.40T + 79T^{2}
83 1+(7.52−13.0i)T+(−41.5−71.8i)T2 1 + (7.52 - 13.0i)T + (-41.5 - 71.8i)T^{2}
89 1+(−4.54−7.86i)T+(−44.5+77.0i)T2 1 + (-4.54 - 7.86i)T + (-44.5 + 77.0i)T^{2}
97 1+(0.949−1.64i)T+(−48.5−84.0i)T2 1 + (0.949 - 1.64i)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

−10.63185656846347446293502046428, −9.680710951458862262947061457276, −9.013694290153013476667875447271, −7.85405655797020329254002825004, −7.11768679756588092783429912513, −5.46928973068733747327219655113, −5.26757087448221937690466599494, −3.80868291453927810432760875015, −1.57707763972397306361969179754, −0.07730527132022176145649915172, 1.92525533308091082059949117905, 4.04189447426622376999870256959, 4.84757211110734950227638390386, 6.05580858432788678072738000084, 6.95276772005982495214741309747, 7.977274873198901111298825283497, 9.102121994355509701997388345539, 10.11362013257539739646320478632, 10.32490200106490671546109271475, 11.42461451895730537777692912459

Graph of the ZZ-function along the critical line