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Properties

Label 2-3549-273.179-c0-0-3
Degree 22
Conductor 35493549
Sign 0.746−0.665i0.746 - 0.665i
Analytic cond. 1.771181.77118
Root an. cond. 1.330851.33085
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)7-s + 9-s + (0.5 + 0.866i)12-s + (−0.499 + 0.866i)16-s + i·19-s + (−0.866 − 0.5i)21-s + (0.5 − 0.866i)25-s + 27-s − 0.999i·28-s + (1.73 + i)31-s + (0.5 + 0.866i)36-s + (0.866 + 0.5i)37-s + (−0.5 + 0.866i)43-s + ⋯
L(s)  = 1  + 3-s + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)7-s + 9-s + (0.5 + 0.866i)12-s + (−0.499 + 0.866i)16-s + i·19-s + (−0.866 − 0.5i)21-s + (0.5 − 0.866i)25-s + 27-s − 0.999i·28-s + (1.73 + i)31-s + (0.5 + 0.866i)36-s + (0.866 + 0.5i)37-s + (−0.5 + 0.866i)43-s + ⋯

Functional equation

Λ(s)=(3549s/2 ΓC(s) L(s)=((0.746−0.665i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3549s/2 ΓC(s) L(s)=((0.746−0.665i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35493549    =    3⋅7⋅1323 \cdot 7 \cdot 13^{2}
Sign: 0.746−0.665i0.746 - 0.665i
Analytic conductor: 1.771181.77118
Root analytic conductor: 1.330851.33085
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3549(1544,⋅)\chi_{3549} (1544, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3549, ( :0), 0.746−0.665i)(2,\ 3549,\ (\ :0),\ 0.746 - 0.665i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.9356285311.935628531
L(12)L(\frac12) ≈\approx 1.9356285311.935628531
L(1)L(1) 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−T 1 - T
7 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
13 1 1
good2 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
5 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
11 1+T2 1 + T^{2}
17 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
19 1−iT−T2 1 - iT - T^{2}
23 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1+(−1.73−i)T+(0.5+0.866i)T2 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2}
37 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
41 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
43 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
47 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
53 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
59 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
61 1+T+T2 1 + T + T^{2}
67 1+2iT−T2 1 + 2iT - T^{2}
71 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
73 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
79 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
97 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)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

−8.644127809115248737171812593705, −8.059554450740223429182290489657, −7.53626654274057198144173082526, −6.59999241918681510899941237626, −6.30943569768828270496703856253, −4.68840220931324339090616997439, −4.01253049385655196568964563247, −3.14141736534737043264376439409, −2.73761401666886460397118467676, −1.47408669519642256485814320264, 1.10542088073475575595822420804, 2.42759083596634540983900877413, 2.78059953179233360832616056212, 3.90335818972712046814950259395, 4.87413102164184817888939734930, 5.73491589312268813654177033296, 6.59380962106653411126765481443, 7.04097011270165066588540874384, 7.943966334017700608655122584777, 8.838940976519122052088952844292

Graph of the ZZ-function along the critical line