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Properties

Label 2-2160-180.79-c0-0-1
Degree 22
Conductor 21602160
Sign −0.173+0.984i-0.173 + 0.984i
Analytic cond. 1.077981.07798
Root an. cond. 1.038251.03825
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)5-s + (0.866 − 1.5i)7-s + (−0.866 − 1.5i)23-s + (−0.499 + 0.866i)25-s + (−0.5 + 0.866i)29-s − 1.73·35-s + (0.5 + 0.866i)41-s + (0.866 − 1.5i)47-s + (−1 − 1.73i)49-s + (−0.5 + 0.866i)61-s + (−0.866 − 1.5i)67-s + (−0.866 + 1.5i)83-s + 89-s + (1 − 1.73i)101-s + 1.73·107-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + (0.866 − 1.5i)7-s + (−0.866 − 1.5i)23-s + (−0.499 + 0.866i)25-s + (−0.5 + 0.866i)29-s − 1.73·35-s + (0.5 + 0.866i)41-s + (0.866 − 1.5i)47-s + (−1 − 1.73i)49-s + (−0.5 + 0.866i)61-s + (−0.866 − 1.5i)67-s + (−0.866 + 1.5i)83-s + 89-s + (1 − 1.73i)101-s + 1.73·107-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((−0.173+0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s) L(s)=((−0.173+0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: −0.173+0.984i-0.173 + 0.984i
Analytic conductor: 1.077981.07798
Root analytic conductor: 1.038251.03825
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2160(559,⋅)\chi_{2160} (559, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :0), −0.173+0.984i)(2,\ 2160,\ (\ :0),\ -0.173 + 0.984i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0649312031.064931203
L(12)L(\frac12) ≈\approx 1.0649312031.064931203
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 1
3 1 1
5 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good7 1+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
13 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
17 1−T2 1 - T^{2}
19 1−T2 1 - T^{2}
23 1+(0.866+1.5i)T+(−0.5+0.866i)T2 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2}
29 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1−T2 1 - T^{2}
41 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
43 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
47 1+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2}
53 1−T2 1 - T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)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+(0.866+1.5i)T+(−0.5+0.866i)T2 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1−T2 1 - T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+(0.866−1.5i)T+(−0.5−0.866i)T2 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2}
89 1−T+T2 1 - T + T^{2}
97 1+(0.5+0.866i)T2 1 + (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

−8.844349915268557763908733644449, −8.250418633241100987435494834928, −7.56958714984671079574676366834, −6.94358138324772474405549435297, −5.80882538162803068541835881723, −4.72765627252375934363096693239, −4.35368246118398338684239370523, −3.47721463976858957705235265307, −1.88191527200502079658863358174, −0.76524831502332005118360907554, 1.85507581649513027495424109356, 2.66039398739313116975375322202, 3.68691545613280419740794780920, 4.66258351014817769349426701055, 5.73698672350256743905285147228, 6.10152030812417905127427122256, 7.43865526607474839635750959135, 7.78655611514018007652620768231, 8.697566198040515126857762338451, 9.368157754533704897246694691815

Graph of the ZZ-function along the critical line