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Properties

Label 2-3549-21.2-c0-0-6
Degree 22
Conductor 35493549
Sign 0.997−0.0633i0.997 - 0.0633i
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  + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.866 + 0.5i)5-s + 0.999·6-s + (−0.5 + 0.866i)7-s − i·8-s + (0.499 − 0.866i)9-s + (0.499 + 0.866i)10-s + (−0.866 + 0.499i)14-s + 0.999·15-s + (0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (0.866 − 0.499i)18-s + 0.999i·21-s + (−0.866 − 0.5i)23-s + (−0.500 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.866 + 0.5i)5-s + 0.999·6-s + (−0.5 + 0.866i)7-s − i·8-s + (0.499 − 0.866i)9-s + (0.499 + 0.866i)10-s + (−0.866 + 0.499i)14-s + 0.999·15-s + (0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (0.866 − 0.499i)18-s + 0.999i·21-s + (−0.866 − 0.5i)23-s + (−0.500 − 0.866i)24-s + ⋯

Functional equation

Λ(s)=(3549s/2 ΓC(s) L(s)=((0.997−0.0633i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3549s/2 ΓC(s) L(s)=((0.997−0.0633i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35493549    =    3⋅7⋅1323 \cdot 7 \cdot 13^{2}
Sign: 0.997−0.0633i0.997 - 0.0633i
Analytic conductor: 1.771181.77118
Root analytic conductor: 1.330851.33085
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3549(170,⋅)\chi_{3549} (170, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3549, ( :0), 0.997−0.0633i)(2,\ 3549,\ (\ :0),\ 0.997 - 0.0633i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.8750394952.875039495
L(12)L(\frac12) ≈\approx 2.8750394952.875039495
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+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
7 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
13 1 1
good2 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
5 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
17 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
19 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
23 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
29 1−iT−T2 1 - iT - T^{2}
31 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
37 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
41 1+iT−T2 1 + iT - T^{2}
43 1−T+T2 1 - T + T^{2}
47 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
53 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
59 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
61 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
67 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
71 1−iT−T2 1 - iT - T^{2}
73 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
79 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
97 1+T+T2 1 + T + 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

−8.843098293354370364654632904539, −7.86143824559818728660054651268, −7.01079897730449712325292416779, −6.42933274499263184945330649007, −5.87209672797094643549803054868, −5.16822631286312359879695882913, −4.08736645078950007892335390638, −3.13063636244558393235943834579, −2.57439242690243099995285761355, −1.40031750657258055488451401178, 1.58650792027432124703707891705, 2.46360901423523538128158775844, 3.40983729907087796069023361136, 3.98956672634286970689886381270, 4.64067466894784541431617508837, 5.56522204954389770241310280453, 6.17991199966726997296602831919, 7.58347333446950415766391377478, 7.913524147935687966006768327190, 8.865495134880672236171595117142

Graph of the ZZ-function along the critical line