↑

Properties

Label 2-5712-1.1-c1-0-3
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 − 2.29·5-s − 7-s + 9-s − 4.29·11-s − 3.87·13-s − 2.29·15-s − 17-s − 3.08·19-s − 21-s + 3.08·23-s + 0.276·25-s + 27-s + 6·29-s + 6.16·31-s − 4.29·33-s + 2.29·35-s + 0.786·37-s − 3.87·39-s − 1.08·41-s − 5.87·43-s − 2.29·45-s − 6.16·47-s + 49-s − 51-s − 2.95·53-s + 9.87·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.02·5-s − 0.377·7-s + 0.333·9-s − 1.29·11-s − 1.07·13-s − 0.593·15-s − 0.242·17-s − 0.707·19-s − 0.218·21-s + 0.643·23-s + 0.0553·25-s + 0.192·27-s + 1.11·29-s + 1.10·31-s − 0.748·33-s + 0.388·35-s + 0.129·37-s − 0.619·39-s − 0.169·41-s − 0.895·43-s − 0.342·45-s − 0.899·47-s + 0.142·49-s − 0.140·51-s − 0.405·53-s + 1.33·55-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 1.0727719131.072771913
L(12)L(\frac12) ≈\approx 1.0727719131.072771913
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+2.29T+5T2 1 + 2.29T + 5T^{2}
11 1+4.29T+11T2 1 + 4.29T + 11T^{2}
13 1+3.87T+13T2 1 + 3.87T + 13T^{2}
19 1+3.08T+19T2 1 + 3.08T + 19T^{2}
23 1−3.08T+23T2 1 - 3.08T + 23T^{2}
29 1−6T+29T2 1 - 6T + 29T^{2}
31 1−6.16T+31T2 1 - 6.16T + 31T^{2}
37 1−0.786T+37T2 1 - 0.786T + 37T^{2}
41 1+1.08T+41T2 1 + 1.08T + 41T^{2}
43 1+5.87T+43T2 1 + 5.87T + 43T^{2}
47 1+6.16T+47T2 1 + 6.16T + 47T^{2}
53 1+2.95T+53T2 1 + 2.95T + 53T^{2}
59 1−7.02T+59T2 1 - 7.02T + 59T^{2}
61 1+5.02T+61T2 1 + 5.02T + 61T^{2}
67 1+4.95T+67T2 1 + 4.95T + 67T^{2}
71 1+4T+71T2 1 + 4T + 71T^{2}
73 1+1.38T+73T2 1 + 1.38T + 73T^{2}
79 1−2.78T+79T2 1 - 2.78T + 79T^{2}
83 1−4.36T+83T2 1 - 4.36T + 83T^{2}
89 1−15.5T+89T2 1 - 15.5T + 89T^{2}
97 1−17.1T+97T2 1 - 17.1T + 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

−8.089741203578080816260973777388, −7.55563241897776195663107974392, −6.89260712453926549664717789150, −6.10822121568372702479257103101, −4.79805057991774791586804124146, −4.69395634370256412208224015291, −3.49915208998746484028128010957, −2.88347758299567771910933028806, −2.12218887990436832979342738792, −0.50362081728573779615504243150, 0.50362081728573779615504243150, 2.12218887990436832979342738792, 2.88347758299567771910933028806, 3.49915208998746484028128010957, 4.69395634370256412208224015291, 4.79805057991774791586804124146, 6.10822121568372702479257103101, 6.89260712453926549664717789150, 7.55563241897776195663107974392, 8.089741203578080816260973777388

Graph of the ZZ-function along the critical line