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Properties

Label 2-546-13.3-c1-0-3
Degree 22
Conductor 546546
Sign 0.342−0.939i0.342 - 0.939i
Analytic cond. 4.359834.35983
Root an. cond. 2.088022.08802
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  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s − 2·5-s + (0.499 − 0.866i)6-s + (0.5 − 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (1 + 1.73i)10-s + (2.58 + 4.48i)11-s − 0.999·12-s + (−1.5 − 3.27i)13-s − 0.999·14-s + (−1 − 1.73i)15-s + (−0.5 − 0.866i)16-s + (−2.5 + 4.33i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.894·5-s + (0.204 − 0.353i)6-s + (0.188 − 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.316 + 0.547i)10-s + (0.780 + 1.35i)11-s − 0.288·12-s + (−0.416 − 0.909i)13-s − 0.267·14-s + (−0.258 − 0.447i)15-s + (−0.125 − 0.216i)16-s + (−0.606 + 1.05i)17-s + ⋯

Functional equation

Λ(s)=(546s/2 ΓC(s) L(s)=((0.342−0.939i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(546s/2 ΓC(s+1/2) L(s)=((0.342−0.939i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 546546    =    2⋅3⋅7⋅132 \cdot 3 \cdot 7 \cdot 13
Sign: 0.342−0.939i0.342 - 0.939i
Analytic conductor: 4.359834.35983
Root analytic conductor: 2.088022.08802
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ546(211,⋅)\chi_{546} (211, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 546, ( :1/2), 0.342−0.939i)(2,\ 546,\ (\ :1/2),\ 0.342 - 0.939i)

Particular Values

L(1)L(1) ≈\approx 0.734396+0.513764i0.734396 + 0.513764i
L(12)L(\frac12) ≈\approx 0.734396+0.513764i0.734396 + 0.513764i
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)
bad2 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
3 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
7 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
13 1+(1.5+3.27i)T 1 + (1.5 + 3.27i)T
good5 1+2T+5T2 1 + 2T + 5T^{2}
11 1+(−2.58−4.48i)T+(−5.5+9.52i)T2 1 + (-2.58 - 4.48i)T + (-5.5 + 9.52i)T^{2}
17 1+(2.5−4.33i)T+(−8.5−14.7i)T2 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.58−2.75i)T+(−9.5−16.4i)T2 1 + (1.58 - 2.75i)T + (-9.5 - 16.4i)T^{2}
23 1+(−4.08−7.08i)T+(−11.5+19.9i)T2 1 + (-4.08 - 7.08i)T + (-11.5 + 19.9i)T^{2}
29 1+(−1.58−2.75i)T+(−14.5+25.1i)T2 1 + (-1.58 - 2.75i)T + (-14.5 + 25.1i)T^{2}
31 1−4.17T+31T2 1 - 4.17T + 31T^{2}
37 1+(1+1.73i)T+(−18.5+32.0i)T2 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2}
41 1+(−3.58−6.21i)T+(−20.5+35.5i)T2 1 + (-3.58 - 6.21i)T + (-20.5 + 35.5i)T^{2}
43 1+(6.08−10.5i)T+(−21.5−37.2i)T2 1 + (6.08 - 10.5i)T + (-21.5 - 37.2i)T^{2}
47 1+7.17T+47T2 1 + 7.17T + 47T^{2}
53 1+3T+53T2 1 + 3T + 53T^{2}
59 1+(−6.08+10.5i)T+(−29.5−51.0i)T2 1 + (-6.08 + 10.5i)T + (-29.5 - 51.0i)T^{2}
61 1+(−1.5+2.59i)T+(−30.5−52.8i)T2 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1.91−3.30i)T+(−33.5+58.0i)T2 1 + (-1.91 - 3.30i)T + (-33.5 + 58.0i)T^{2}
71 1+(−1.08+1.88i)T+(−35.5−61.4i)T2 1 + (-1.08 + 1.88i)T + (-35.5 - 61.4i)T^{2}
73 1−4T+73T2 1 - 4T + 73T^{2}
79 1−13.1T+79T2 1 - 13.1T + 79T^{2}
83 1−6.17T+83T2 1 - 6.17T + 83T^{2}
89 1+(4.5+7.79i)T+(−44.5+77.0i)T2 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2}
97 1+(−5.17+8.97i)T+(−48.5−84.0i)T2 1 + (-5.17 + 8.97i)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

−10.97824405833601616701795341155, −10.00230279821300597157895990156, −9.475998439311390433559590264202, −8.240579418026804574474824537766, −7.76659180068789645909828552232, −6.64646963877933473075456209287, −4.94494751860031872216306115013, −4.11893531363589770482950566249, −3.26423483179434402926065797716, −1.66152350960272933531137111645, 0.58423465200281592779347530177, 2.51635319766143103932682499058, 3.97482456923709664462515045886, 5.04419061149606610582208603161, 6.49743424848565371082611113071, 6.90302359436369252990671801839, 8.066906542196925941207420155090, 8.736498086756324509579271754328, 9.278858989398728013863616779979, 10.73919597840634787213072779463

Graph of the ZZ-function along the critical line