↑

Properties

Label 2-1759-1.1-c1-0-130
Degree 22
Conductor 17591759
Sign −1-1
Analytic cond. 14.045614.0456
Root an. cond. 3.747753.74775
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.99·2-s − 0.531·3-s + 1.99·4-s + 0.113·5-s − 1.06·6-s − 3.12·7-s − 0.00957·8-s − 2.71·9-s + 0.227·10-s + 2.02·11-s − 1.06·12-s + 5.47·13-s − 6.25·14-s − 0.0604·15-s − 4.00·16-s − 4.25·17-s − 5.43·18-s − 1.81·19-s + 0.227·20-s + 1.66·21-s + 4.04·22-s − 1.75·23-s + 0.00508·24-s − 4.98·25-s + 10.9·26-s + 3.03·27-s − 6.24·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.306·3-s + 0.997·4-s + 0.0508·5-s − 0.433·6-s − 1.18·7-s − 0.00338·8-s − 0.905·9-s + 0.0719·10-s + 0.610·11-s − 0.306·12-s + 1.51·13-s − 1.67·14-s − 0.0156·15-s − 1.00·16-s − 1.03·17-s − 1.28·18-s − 0.416·19-s + 0.0507·20-s + 0.362·21-s + 0.862·22-s − 0.365·23-s + 0.00103·24-s − 0.997·25-s + 2.14·26-s + 0.584·27-s − 1.17·28-s + ⋯

Functional equation

Λ(s)=(1759s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1759 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1759s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1759 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17591759
Sign: −1-1
Analytic conductor: 14.045614.0456
Root analytic conductor: 3.747753.74775
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1759, ( :1/2), −1)(2,\ 1759,\ (\ :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)
bad1759 1+T 1 + T
good2 1−1.99T+2T2 1 - 1.99T + 2T^{2}
3 1+0.531T+3T2 1 + 0.531T + 3T^{2}
5 1−0.113T+5T2 1 - 0.113T + 5T^{2}
7 1+3.12T+7T2 1 + 3.12T + 7T^{2}
11 1−2.02T+11T2 1 - 2.02T + 11T^{2}
13 1−5.47T+13T2 1 - 5.47T + 13T^{2}
17 1+4.25T+17T2 1 + 4.25T + 17T^{2}
19 1+1.81T+19T2 1 + 1.81T + 19T^{2}
23 1+1.75T+23T2 1 + 1.75T + 23T^{2}
29 1+4.32T+29T2 1 + 4.32T + 29T^{2}
31 1+3.99T+31T2 1 + 3.99T + 31T^{2}
37 1−10.2T+37T2 1 - 10.2T + 37T^{2}
41 1+12.2T+41T2 1 + 12.2T + 41T^{2}
43 1+4.14T+43T2 1 + 4.14T + 43T^{2}
47 1+5.02T+47T2 1 + 5.02T + 47T^{2}
53 1+11.0T+53T2 1 + 11.0T + 53T^{2}
59 1−9.61T+59T2 1 - 9.61T + 59T^{2}
61 1+10.2T+61T2 1 + 10.2T + 61T^{2}
67 1+6.53T+67T2 1 + 6.53T + 67T^{2}
71 1−1.16T+71T2 1 - 1.16T + 71T^{2}
73 1−14.5T+73T2 1 - 14.5T + 73T^{2}
79 1−12.2T+79T2 1 - 12.2T + 79T^{2}
83 1−3.03T+83T2 1 - 3.03T + 83T^{2}
89 1−12.8T+89T2 1 - 12.8T + 89T^{2}
97 1−5.16T+97T2 1 - 5.16T + 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

−9.005466146516908656650730449311, −8.129619172262496295196296369148, −6.64693813202972202119046682512, −6.30603728893746138703646411572, −5.80255927857673420319978561557, −4.75275692530387377372957102853, −3.70201967475401659036207751157, −3.35455475132659221496475241587, −2.06636640658925882430744945313, 0, 2.06636640658925882430744945313, 3.35455475132659221496475241587, 3.70201967475401659036207751157, 4.75275692530387377372957102853, 5.80255927857673420319978561557, 6.30603728893746138703646411572, 6.64693813202972202119046682512, 8.129619172262496295196296369148, 9.005466146516908656650730449311

Graph of the ZZ-function along the critical line