↑

Properties

Label 2-5712-1.1-c1-0-16
Degree 22
Conductor 57125712
Sign 11
Analytic cond. 45.610545.6105
Root an. cond. 6.753556.75355
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 0.561·5-s + 7-s + 9-s − 2.56·11-s + 4.56·13-s + 0.561·15-s + 17-s + 0.561·19-s − 21-s − 2.56·23-s − 4.68·25-s − 27-s − 1.12·29-s + 5.12·31-s + 2.56·33-s − 0.561·35-s + 7.12·37-s − 4.56·39-s + 4.56·41-s − 1.43·43-s − 0.561·45-s − 5.12·47-s + 49-s − 51-s + 6·53-s + 1.43·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.251·5-s + 0.377·7-s + 0.333·9-s − 0.772·11-s + 1.26·13-s + 0.144·15-s + 0.242·17-s + 0.128·19-s − 0.218·21-s − 0.534·23-s − 0.936·25-s − 0.192·27-s − 0.208·29-s + 0.920·31-s + 0.445·33-s − 0.0949·35-s + 1.17·37-s − 0.730·39-s + 0.712·41-s − 0.219·43-s − 0.0837·45-s − 0.747·47-s + 0.142·49-s − 0.140·51-s + 0.824·53-s + 0.193·55-s + ⋯

Functional equation

Λ(s)=(5712s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5712s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57125712    =    24⋅3⋅7⋅172^{4} \cdot 3 \cdot 7 \cdot 17
Sign: 11
Analytic conductor: 45.610545.6105
Root analytic conductor: 6.753556.75355
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5712, ( :1/2), 1)(2,\ 5712,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4859377801.485937780
L(12)L(\frac12) ≈\approx 1.4859377801.485937780
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
3 1+T 1 + T
7 1−T 1 - T
17 1−T 1 - T
good5 1+0.561T+5T2 1 + 0.561T + 5T^{2}
11 1+2.56T+11T2 1 + 2.56T + 11T^{2}
13 1−4.56T+13T2 1 - 4.56T + 13T^{2}
19 1−0.561T+19T2 1 - 0.561T + 19T^{2}
23 1+2.56T+23T2 1 + 2.56T + 23T^{2}
29 1+1.12T+29T2 1 + 1.12T + 29T^{2}
31 1−5.12T+31T2 1 - 5.12T + 31T^{2}
37 1−7.12T+37T2 1 - 7.12T + 37T^{2}
41 1−4.56T+41T2 1 - 4.56T + 41T^{2}
43 1+1.43T+43T2 1 + 1.43T + 43T^{2}
47 1+5.12T+47T2 1 + 5.12T + 47T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1+11.3T+59T2 1 + 11.3T + 59T^{2}
61 1−14.2T+61T2 1 - 14.2T + 61T^{2}
67 1−4T+67T2 1 - 4T + 67T^{2}
71 1+4T+71T2 1 + 4T + 71T^{2}
73 1+5.12T+73T2 1 + 5.12T + 73T^{2}
79 1+7.36T+79T2 1 + 7.36T + 79T^{2}
83 1−5.12T+83T2 1 - 5.12T + 83T^{2}
89 1−13.3T+89T2 1 - 13.3T + 89T^{2}
97 1+14.2T+97T2 1 + 14.2T + 97T^{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

−8.017119626327761558067972188882, −7.57261570996245481318485659009, −6.56916198784719116514440158298, −5.93211149174073308880979943336, −5.36413881549599092246633704766, −4.44045270636198190629524015318, −3.83252769944893195986423059417, −2.81621326117819935927835042674, −1.73392163390563410545671193219, −0.68245589433683787157343506089, 0.68245589433683787157343506089, 1.73392163390563410545671193219, 2.81621326117819935927835042674, 3.83252769944893195986423059417, 4.44045270636198190629524015318, 5.36413881549599092246633704766, 5.93211149174073308880979943336, 6.56916198784719116514440158298, 7.57261570996245481318485659009, 8.017119626327761558067972188882

Graph of the ZZ-function along the critical line