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Properties

Label 2-2156-308.163-c0-0-0
Degree 22
Conductor 21562156
Sign 0.0809−0.996i0.0809 - 0.996i
Analytic cond. 1.075981.07598
Root an. cond. 1.037291.03729
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (0.669 − 0.743i)11-s + (0.913 − 0.406i)16-s + (−0.669 + 0.743i)18-s + (0.809 + 0.587i)22-s + (1.01 + 0.587i)23-s + (−0.913 − 0.406i)25-s + (0.5 + 0.363i)29-s + (0.499 + 0.866i)32-s + (−0.809 − 0.587i)36-s + (1.47 − 0.658i)37-s + 1.90i·43-s + (−0.5 + 0.866i)44-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (0.669 − 0.743i)11-s + (0.913 − 0.406i)16-s + (−0.669 + 0.743i)18-s + (0.809 + 0.587i)22-s + (1.01 + 0.587i)23-s + (−0.913 − 0.406i)25-s + (0.5 + 0.363i)29-s + (0.499 + 0.866i)32-s + (−0.809 − 0.587i)36-s + (1.47 − 0.658i)37-s + 1.90i·43-s + (−0.5 + 0.866i)44-s + ⋯

Functional equation

Λ(s)=(2156s/2 ΓC(s) L(s)=((0.0809−0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0809 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2156s/2 ΓC(s) L(s)=((0.0809−0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0809 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21562156    =    22⋅72⋅112^{2} \cdot 7^{2} \cdot 11
Sign: 0.0809−0.996i0.0809 - 0.996i
Analytic conductor: 1.075981.07598
Root analytic conductor: 1.037291.03729
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2156(471,⋅)\chi_{2156} (471, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2156, ( :0), 0.0809−0.996i)(2,\ 2156,\ (\ :0),\ 0.0809 - 0.996i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2143502001.214350200
L(12)L(\frac12) ≈\approx 1.2143502001.214350200
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+(−0.104−0.994i)T 1 + (-0.104 - 0.994i)T
7 1 1
11 1+(−0.669+0.743i)T 1 + (-0.669 + 0.743i)T
good3 1+(−0.669−0.743i)T2 1 + (-0.669 - 0.743i)T^{2}
5 1+(0.913+0.406i)T2 1 + (0.913 + 0.406i)T^{2}
13 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
17 1+(−0.104−0.994i)T2 1 + (-0.104 - 0.994i)T^{2}
19 1+(0.978−0.207i)T2 1 + (0.978 - 0.207i)T^{2}
23 1+(−1.01−0.587i)T+(0.5+0.866i)T2 1 + (-1.01 - 0.587i)T + (0.5 + 0.866i)T^{2}
29 1+(−0.5−0.363i)T+(0.309+0.951i)T2 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2}
31 1+(−0.913+0.406i)T2 1 + (-0.913 + 0.406i)T^{2}
37 1+(−1.47+0.658i)T+(0.669−0.743i)T2 1 + (-1.47 + 0.658i)T + (0.669 - 0.743i)T^{2}
41 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
43 1−1.90iT−T2 1 - 1.90iT - T^{2}
47 1+(0.978−0.207i)T2 1 + (0.978 - 0.207i)T^{2}
53 1+(0.604−0.128i)T+(0.913−0.406i)T2 1 + (0.604 - 0.128i)T + (0.913 - 0.406i)T^{2}
59 1+(0.978+0.207i)T2 1 + (0.978 + 0.207i)T^{2}
61 1+(0.913+0.406i)T2 1 + (0.913 + 0.406i)T^{2}
67 1+(−1.64+0.951i)T+(0.5−0.866i)T2 1 + (-1.64 + 0.951i)T + (0.5 - 0.866i)T^{2}
71 1+(1.80−0.587i)T+(0.809−0.587i)T2 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2}
73 1+(−0.978−0.207i)T2 1 + (-0.978 - 0.207i)T^{2}
79 1+(1.41−1.27i)T+(0.104−0.994i)T2 1 + (1.41 - 1.27i)T + (0.104 - 0.994i)T^{2}
83 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
89 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
97 1+(−0.809−0.587i)T2 1 + (-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

−9.390653055162438620885447152231, −8.468255155911914041637762691564, −7.85032774459905379509427940607, −7.13234202738077366663076337299, −6.33076629989246699792657203148, −5.61966512870565069430697288674, −4.68861280145512400422675370761, −4.01531205183555088573929775424, −2.92486189706581656547484522016, −1.26683188501870678667768599857, 1.06641710939528831820378112154, 2.13443325513619328430441501857, 3.26943665264599669528267914914, 4.14003316636666484513030518336, 4.71928452536519784101442510073, 5.84010297581943704800357386795, 6.72695083850832415561837108611, 7.56290744372002735875204274828, 8.627010823943086648310666178849, 9.270622441915439542975924100235

Graph of the ZZ-function along the critical line