↑

Properties

Label 2-5445-1.1-c1-0-133
Degree 22
Conductor 54455445
Sign −1-1
Analytic cond. 43.478543.4785
Root an. cond. 6.593826.59382
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 5-s − 3·8-s − 10-s − 2·13-s − 16-s + 6·17-s + 4·19-s + 20-s − 4·23-s + 25-s − 2·26-s + 6·29-s − 8·31-s + 5·32-s + 6·34-s − 2·37-s + 4·38-s + 3·40-s + 2·41-s − 4·43-s − 4·46-s + 12·47-s − 7·49-s + 50-s + 2·52-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.06·8-s − 0.316·10-s − 0.554·13-s − 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.223·20-s − 0.834·23-s + 1/5·25-s − 0.392·26-s + 1.11·29-s − 1.43·31-s + 0.883·32-s + 1.02·34-s − 0.328·37-s + 0.648·38-s + 0.474·40-s + 0.312·41-s − 0.609·43-s − 0.589·46-s + 1.75·47-s − 49-s + 0.141·50-s + 0.277·52-s + ⋯

Functional equation

Λ(s)=(5445s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(5445s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 54455445    =    32⋅5⋅1123^{2} \cdot 5 \cdot 11^{2}
Sign: −1-1
Analytic conductor: 43.478543.4785
Root analytic conductor: 6.593826.59382
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 5445, ( :1/2), −1)(2,\ 5445,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad3 1 1
5 1+T 1 + T
11 1 1
good2 1−T+pT2 1 - T + p T^{2}
7 1+pT2 1 + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+16T+pT2 1 + 16 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1+14T+pT2 1 + 14 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1−10T+pT2 1 - 10 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

−7.65244290168268394743634881542, −7.25611852136541238398716522907, −6.08693701335829428157980562583, −5.54640979880007283696239996782, −4.87806253960044967314702320131, −4.10497602187160877381582357369, −3.40349928139289108990558562724, −2.73199597836315214499399961863, −1.27797148118613320465803316120, 0, 1.27797148118613320465803316120, 2.73199597836315214499399961863, 3.40349928139289108990558562724, 4.10497602187160877381582357369, 4.87806253960044967314702320131, 5.54640979880007283696239996782, 6.08693701335829428157980562583, 7.25611852136541238398716522907, 7.65244290168268394743634881542

Graph of the ZZ-function along the critical line