↑

Properties

Label 2-1083-1.1-c1-0-56
Degree 22
Conductor 10831083
Sign −1-1
Analytic cond. 8.647798.64779
Root an. cond. 2.940712.94071
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  + 2·2-s + 3-s + 2·4-s − 3·5-s + 2·6-s − 5·7-s + 9-s − 6·10-s + 11-s + 2·12-s − 2·13-s − 10·14-s − 3·15-s − 4·16-s − 17-s + 2·18-s − 6·20-s − 5·21-s + 2·22-s − 4·23-s + 4·25-s − 4·26-s + 27-s − 10·28-s + 2·29-s − 6·30-s + 6·31-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s − 1.34·5-s + 0.816·6-s − 1.88·7-s + 1/3·9-s − 1.89·10-s + 0.301·11-s + 0.577·12-s − 0.554·13-s − 2.67·14-s − 0.774·15-s − 16-s − 0.242·17-s + 0.471·18-s − 1.34·20-s − 1.09·21-s + 0.426·22-s − 0.834·23-s + 4/5·25-s − 0.784·26-s + 0.192·27-s − 1.88·28-s + 0.371·29-s − 1.09·30-s + 1.07·31-s + ⋯

Functional equation

Λ(s)=(1083s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1083s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10831083    =    3⋅1923 \cdot 19^{2}
Sign: −1-1
Analytic conductor: 8.647798.64779
Root analytic conductor: 2.940712.94071
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1083, ( :1/2), −1)(2,\ 1083,\ (\ :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−T 1 - T
19 1 1
good2 1−pT+pT2 1 - p T + p T^{2}
5 1+3T+pT2 1 + 3 T + p T^{2}
7 1+5T+pT2 1 + 5 T + p T^{2}
11 1−T+pT2 1 - T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+T+pT2 1 + T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1−6T+pT2 1 - 6 T + p T^{2}
37 1+pT2 1 + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1+9T+pT2 1 + 9 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1+T+pT2 1 + T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+11T+pT2 1 + 11 T + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−6T+pT2 1 - 6 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

−9.503827882841026216201178111482, −8.598738845501152801902892747277, −7.58489736631665346168199395971, −6.70211043109012358161209293170, −6.16139530761373947123603078754, −4.78903745519092577014884991606, −3.96503328962323463176155683264, −3.37953587462828366448462440771, −2.64121548040677802901277139388, 0, 2.64121548040677802901277139388, 3.37953587462828366448462440771, 3.96503328962323463176155683264, 4.78903745519092577014884991606, 6.16139530761373947123603078754, 6.70211043109012358161209293170, 7.58489736631665346168199395971, 8.598738845501152801902892747277, 9.503827882841026216201178111482

Graph of the ZZ-function along the critical line