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Properties

Label 2-3549-273.107-c0-0-2
Degree 22
Conductor 35493549
Sign 0.795+0.606i0.795 + 0.606i
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  + (0.5 − 0.866i)3-s + 4-s + (0.866 + 0.5i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)12-s + 16-s + (0.866 + 1.5i)19-s + (0.866 − 0.499i)21-s + (−0.5 − 0.866i)25-s − 0.999·27-s + (0.866 + 0.5i)28-s + (−0.499 − 0.866i)36-s − 1.73·37-s + (−0.5 + 0.866i)43-s + (0.5 − 0.866i)48-s + (0.499 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + 4-s + (0.866 + 0.5i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)12-s + 16-s + (0.866 + 1.5i)19-s + (0.866 − 0.499i)21-s + (−0.5 − 0.866i)25-s − 0.999·27-s + (0.866 + 0.5i)28-s + (−0.499 − 0.866i)36-s − 1.73·37-s + (−0.5 + 0.866i)43-s + (0.5 − 0.866i)48-s + (0.499 + 0.866i)49-s + ⋯

Functional equation

Λ(s)=(3549s/2 ΓC(s) L(s)=((0.795+0.606i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3549s/2 ΓC(s) L(s)=((0.795+0.606i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35493549    =    3⋅7⋅1323 \cdot 7 \cdot 13^{2}
Sign: 0.795+0.606i0.795 + 0.606i
Analytic conductor: 1.771181.77118
Root analytic conductor: 1.330851.33085
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3549(653,⋅)\chi_{3549} (653, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3549, ( :0), 0.795+0.606i)(2,\ 3549,\ (\ :0),\ 0.795 + 0.606i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.1178705302.117870530
L(12)L(\frac12) ≈\approx 2.1178705302.117870530
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
7 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
13 1 1
good2 1−T2 1 - T^{2}
5 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1−T2 1 - T^{2}
19 1+(−0.866−1.5i)T+(−0.5+0.866i)T2 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2}
23 1−T2 1 - T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
37 1+1.73T+T2 1 + 1.73T + 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−T2 1 - T^{2}
61 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
71 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
73 1+(0.866+1.5i)T+(−0.5+0.866i)T2 1 + (0.866 + 1.5i)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−T2 1 - T^{2}
97 1+(0.866−1.5i)T+(−0.5−0.866i)T2 1 + (0.866 - 1.5i)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

−8.341842528308810192943256738778, −7.902633802517985344660599191744, −7.36004614322624536984571652335, −6.43724257513246240896657147336, −5.90163661378742136935363834672, −5.07343058786051468287813564611, −3.72640027700614220744431703369, −2.95093374238459842463127869838, −1.96862948860913271434557356564, −1.44758545457071734627694642464, 1.47581233420672987487139748645, 2.46498373550549570187149213201, 3.30505441354777963907830174650, 4.10490824913838260422917459534, 5.13001649403389599936093179948, 5.52867993822262665495069683566, 6.89457867441687247974530042758, 7.30015289833782051475192831177, 8.097402233640984565259955108212, 8.785988489396381686628353237494

Graph of the ZZ-function along the critical line