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Properties

Label 2-546-273.146-c1-0-11
Degree 22
Conductor 546546
Sign −0.176−0.984i-0.176 - 0.984i
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  + (−0.866 + 0.5i)2-s + (0.866 + 1.5i)3-s + (0.499 − 0.866i)4-s + 3.46·5-s + (−1.5 − 0.866i)6-s + (0.5 + 2.59i)7-s + 0.999i·8-s + (−1.5 + 2.59i)9-s + (−2.99 + 1.73i)10-s + 1.73·12-s + (−3.5 + 0.866i)13-s + (−1.73 − 2i)14-s + (2.99 + 5.19i)15-s + (−0.5 − 0.866i)16-s + (0.866 − 1.5i)17-s − 3i·18-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 + 0.866i)3-s + (0.249 − 0.433i)4-s + 1.54·5-s + (−0.612 − 0.353i)6-s + (0.188 + 0.981i)7-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.948 + 0.547i)10-s + 0.499·12-s + (−0.970 + 0.240i)13-s + (−0.462 − 0.534i)14-s + (0.774 + 1.34i)15-s + (−0.125 − 0.216i)16-s + (0.210 − 0.363i)17-s − 0.707i·18-s + ⋯

Functional equation

Λ(s)=(546s/2 ΓC(s) L(s)=((−0.176−0.984i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.176 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(546s/2 ΓC(s+1/2) L(s)=((−0.176−0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 546546    =    2⋅3⋅7⋅132 \cdot 3 \cdot 7 \cdot 13
Sign: −0.176−0.984i-0.176 - 0.984i
Analytic conductor: 4.359834.35983
Root analytic conductor: 2.088022.08802
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ546(419,⋅)\chi_{546} (419, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 546, ( :1/2), −0.176−0.984i)(2,\ 546,\ (\ :1/2),\ -0.176 - 0.984i)

Particular Values

L(1)L(1) ≈\approx 1.01963+1.21857i1.01963 + 1.21857i
L(12)L(\frac12) ≈\approx 1.01963+1.21857i1.01963 + 1.21857i
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+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
3 1+(−0.866−1.5i)T 1 + (-0.866 - 1.5i)T
7 1+(−0.5−2.59i)T 1 + (-0.5 - 2.59i)T
13 1+(3.5−0.866i)T 1 + (3.5 - 0.866i)T
good5 1−3.46T+5T2 1 - 3.46T + 5T^{2}
11 1+(5.5−9.52i)T2 1 + (5.5 - 9.52i)T^{2}
17 1+(−0.866+1.5i)T+(−8.5−14.7i)T2 1 + (-0.866 + 1.5i)T + (-8.5 - 14.7i)T^{2}
19 1+(9.5+16.4i)T2 1 + (9.5 + 16.4i)T^{2}
23 1+(−2.59+1.5i)T+(11.5−19.9i)T2 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2}
29 1+(−5.19+3i)T+(14.5−25.1i)T2 1 + (-5.19 + 3i)T + (14.5 - 25.1i)T^{2}
31 1−1.73iT−31T2 1 - 1.73iT - 31T^{2}
37 1+(2+3.46i)T+(−18.5+32.0i)T2 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2}
41 1+(3.46+6i)T+(−20.5+35.5i)T2 1 + (3.46 + 6i)T + (-20.5 + 35.5i)T^{2}
43 1+(0.5−0.866i)T+(−21.5−37.2i)T2 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2}
47 1−3.46T+47T2 1 - 3.46T + 47T^{2}
53 1−9iT−53T2 1 - 9iT - 53T^{2}
59 1+(0.866−1.5i)T+(−29.5−51.0i)T2 1 + (0.866 - 1.5i)T + (-29.5 - 51.0i)T^{2}
61 1+(10.5+6.06i)T+(30.5+52.8i)T2 1 + (10.5 + 6.06i)T + (30.5 + 52.8i)T^{2}
67 1+(−3.5−6.06i)T+(−33.5+58.0i)T2 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2}
71 1+(−12.9−7.5i)T+(35.5+61.4i)T2 1 + (-12.9 - 7.5i)T + (35.5 + 61.4i)T^{2}
73 1+13.8iT−73T2 1 + 13.8iT - 73T^{2}
79 1+10T+79T2 1 + 10T + 79T^{2}
83 1−12.1T+83T2 1 - 12.1T + 83T^{2}
89 1+(7.79+13.5i)T+(−44.5+77.0i)T2 1 + (7.79 + 13.5i)T + (-44.5 + 77.0i)T^{2}
97 1+(−15−8.66i)T+(48.5+84.0i)T2 1 + (-15 - 8.66i)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

−10.54507736664622873519811484186, −10.00157847266662845951578550339, −9.166137375182227261810257608064, −8.857629340978120614309195844748, −7.63965228160706084731862686178, −6.38556416898730539464868350404, −5.46333828162002753260487604304, −4.80520563034396675712180874344, −2.81006683163026639849991467091, −2.02662889901884554633845681718, 1.14280338685557149337617833506, 2.16583274808885723291888675758, 3.26784755989072678240482359509, 4.99813462552597712918966364300, 6.29280873905890804867195163966, 7.03171658278538309707453315981, 7.913404351317510659836037220826, 8.858820249398102415042936491161, 9.800556338728787068138559852898, 10.23334184676300509473401872580

Graph of the ZZ-function along the critical line