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Properties

Label 2-2200-88.27-c0-0-5
Degree 22
Conductor 22002200
Sign −0.0457+0.998i-0.0457 + 0.998i
Analytic cond. 1.097941.09794
Root an. cond. 1.047821.04782
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  + (−0.309 − 0.951i)2-s + (0.5 + 0.363i)3-s + (−0.809 + 0.587i)4-s + (0.190 − 0.587i)6-s + (0.809 + 0.587i)8-s + (−0.190 − 0.587i)9-s + (−0.809 − 0.587i)11-s − 0.618·12-s + (0.309 − 0.951i)16-s + (0.5 − 1.53i)17-s + (−0.5 + 0.363i)18-s + (1.30 + 0.951i)19-s + (−0.309 + 0.951i)22-s + (0.190 + 0.587i)24-s + (0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (0.5 + 0.363i)3-s + (−0.809 + 0.587i)4-s + (0.190 − 0.587i)6-s + (0.809 + 0.587i)8-s + (−0.190 − 0.587i)9-s + (−0.809 − 0.587i)11-s − 0.618·12-s + (0.309 − 0.951i)16-s + (0.5 − 1.53i)17-s + (−0.5 + 0.363i)18-s + (1.30 + 0.951i)19-s + (−0.309 + 0.951i)22-s + (0.190 + 0.587i)24-s + (0.309 − 0.951i)27-s + ⋯

Functional equation

Λ(s)=(2200s/2 ΓC(s) L(s)=((−0.0457+0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2200s/2 ΓC(s) L(s)=((−0.0457+0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22002200    =    23⋅52⋅112^{3} \cdot 5^{2} \cdot 11
Sign: −0.0457+0.998i-0.0457 + 0.998i
Analytic conductor: 1.097941.09794
Root analytic conductor: 1.047821.04782
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2200(2051,⋅)\chi_{2200} (2051, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2200, ( :0), −0.0457+0.998i)(2,\ 2200,\ (\ :0),\ -0.0457 + 0.998i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0212465361.021246536
L(12)L(\frac12) ≈\approx 1.0212465361.021246536
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)
bad2 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
5 1 1
11 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
good3 1+(−0.5−0.363i)T+(0.309+0.951i)T2 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2}
7 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
13 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
17 1+(−0.5+1.53i)T+(−0.809−0.587i)T2 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2}
19 1+(−1.30−0.951i)T+(0.309+0.951i)T2 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2}
23 1−T2 1 - T^{2}
29 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
31 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
37 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
41 1+(0.5+0.363i)T+(0.309+0.951i)T2 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2}
43 1+0.618T+T2 1 + 0.618T + T^{2}
47 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
53 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
59 1+(−1.30+0.951i)T+(0.309−0.951i)T2 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2}
61 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
67 1−1.61T+T2 1 - 1.61T + T^{2}
71 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
73 1+(−0.5+0.363i)T+(0.309−0.951i)T2 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2}
79 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
83 1+(0.190−0.587i)T+(−0.809−0.587i)T2 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2}
89 1−0.618T+T2 1 - 0.618T + T^{2}
97 1+(0.190+0.587i)T+(−0.809+0.587i)T2 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)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

−9.247650372337260065120472888611, −8.399367784788583336861022788743, −7.85658230835371947491012282230, −6.93600964844870759520643863493, −5.54759667591958564634008288264, −4.99173590485576578852466093235, −3.68030398588449833708107847547, −3.25244401850014880466607744765, −2.35828107882529096467711899096, −0.822531400857455061362497654972, 1.42380497132462276101063569383, 2.62145598071620995688868743109, 3.82510869626105543069558411739, 4.98593204750475581071293051266, 5.45094109738482224200887376974, 6.50144902097619749598806099721, 7.33019955386800141383572182915, 7.85782329058285093430648930201, 8.428896677191885020698216158774, 9.217282875419131376557873854181

Graph of the ZZ-function along the critical line