↑

Properties

Label 2-832-1.1-c1-0-3
Degree 22
Conductor 832832
Sign 11
Analytic cond. 6.643556.64355
Root an. cond. 2.577502.57750
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 2·7-s − 3·9-s + 2·11-s + 13-s + 6·17-s + 6·19-s + 8·23-s − 25-s − 2·29-s + 10·31-s + 4·35-s + 6·37-s − 6·41-s − 4·43-s + 6·45-s − 2·47-s − 3·49-s − 6·53-s − 4·55-s + 10·59-s + 2·61-s + 6·63-s − 2·65-s − 10·67-s + 10·71-s + 2·73-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.755·7-s − 9-s + 0.603·11-s + 0.277·13-s + 1.45·17-s + 1.37·19-s + 1.66·23-s − 1/5·25-s − 0.371·29-s + 1.79·31-s + 0.676·35-s + 0.986·37-s − 0.937·41-s − 0.609·43-s + 0.894·45-s − 0.291·47-s − 3/7·49-s − 0.824·53-s − 0.539·55-s + 1.30·59-s + 0.256·61-s + 0.755·63-s − 0.248·65-s − 1.22·67-s + 1.18·71-s + 0.234·73-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: 11
Analytic conductor: 6.643556.64355
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 832, ( :1/2), 1)(2,\ 832,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1956938201.195693820
L(12)L(\frac12) ≈\approx 1.1956938201.195693820
L(32)L(\frac{3}{2}) 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
13 1−T 1 - T
good3 1+pT2 1 + p T^{2}
5 1+2T+pT2 1 + 2 T + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1−10T+pT2 1 - 10 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+2T+pT2 1 + 2 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1−10T+pT2 1 - 10 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−2T+pT2 1 - 2 T + p T^{2}
show more
show less
   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

−10.06947661821394918418145315819, −9.420900365687786965896429939417, −8.448574941224861101937906300558, −7.73101944712663475124809392314, −6.78863087763573008439633106061, −5.85741842949549920807230123970, −4.88282077350781318055546355500, −3.47757941400193154474288876226, −3.07083233977996346947903399516, −0.906662202456000794708098813478, 0.906662202456000794708098813478, 3.07083233977996346947903399516, 3.47757941400193154474288876226, 4.88282077350781318055546355500, 5.85741842949549920807230123970, 6.78863087763573008439633106061, 7.73101944712663475124809392314, 8.448574941224861101937906300558, 9.420900365687786965896429939417, 10.06947661821394918418145315819

Graph of the ZZ-function along the critical line