↑

Properties

Label 2-1875-1.1-c1-0-63
Degree 22
Conductor 18751875
Sign −1-1
Analytic cond. 14.971914.9719
Root an. cond. 3.869363.86936
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.12·2-s − 3-s − 0.740·4-s − 1.12·6-s − 1.11·7-s − 3.07·8-s + 9-s + 3.67·11-s + 0.740·12-s + 4.05·13-s − 1.24·14-s − 1.97·16-s − 2.12·17-s + 1.12·18-s − 4.06·19-s + 1.11·21-s + 4.11·22-s − 6.17·23-s + 3.07·24-s + 4.54·26-s − 27-s + 0.824·28-s − 2.25·29-s + 10.0·31-s + 3.93·32-s − 3.67·33-s − 2.38·34-s + ⋯
L(s)  = 1  + 0.793·2-s − 0.577·3-s − 0.370·4-s − 0.458·6-s − 0.420·7-s − 1.08·8-s + 0.333·9-s + 1.10·11-s + 0.213·12-s + 1.12·13-s − 0.334·14-s − 0.492·16-s − 0.514·17-s + 0.264·18-s − 0.931·19-s + 0.243·21-s + 0.878·22-s − 1.28·23-s + 0.627·24-s + 0.891·26-s − 0.192·27-s + 0.155·28-s − 0.419·29-s + 1.80·31-s + 0.696·32-s − 0.638·33-s − 0.408·34-s + ⋯

Functional equation

Λ(s)=(1875s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1875s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18751875    =    3⋅543 \cdot 5^{4}
Sign: −1-1
Analytic conductor: 14.971914.9719
Root analytic conductor: 3.869363.86936
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1875, ( :1/2), −1)(2,\ 1875,\ (\ :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
5 1 1
good2 1−1.12T+2T2 1 - 1.12T + 2T^{2}
7 1+1.11T+7T2 1 + 1.11T + 7T^{2}
11 1−3.67T+11T2 1 - 3.67T + 11T^{2}
13 1−4.05T+13T2 1 - 4.05T + 13T^{2}
17 1+2.12T+17T2 1 + 2.12T + 17T^{2}
19 1+4.06T+19T2 1 + 4.06T + 19T^{2}
23 1+6.17T+23T2 1 + 6.17T + 23T^{2}
29 1+2.25T+29T2 1 + 2.25T + 29T^{2}
31 1−10.0T+31T2 1 - 10.0T + 31T^{2}
37 1+7.37T+37T2 1 + 7.37T + 37T^{2}
41 1+7.47T+41T2 1 + 7.47T + 41T^{2}
43 1+9.24T+43T2 1 + 9.24T + 43T^{2}
47 1−3.12T+47T2 1 - 3.12T + 47T^{2}
53 1−3.50T+53T2 1 - 3.50T + 53T^{2}
59 1+6.59T+59T2 1 + 6.59T + 59T^{2}
61 1+9.10T+61T2 1 + 9.10T + 61T^{2}
67 1+2.62T+67T2 1 + 2.62T + 67T^{2}
71 1−0.660T+71T2 1 - 0.660T + 71T^{2}
73 1+7.47T+73T2 1 + 7.47T + 73T^{2}
79 1−8.53T+79T2 1 - 8.53T + 79T^{2}
83 1+12.2T+83T2 1 + 12.2T + 83T^{2}
89 1+15.2T+89T2 1 + 15.2T + 89T^{2}
97 1+13.4T+97T2 1 + 13.4T + 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

−8.753735690020116685091553768623, −8.285098339094794154450601857094, −6.72652405724055332287572756453, −6.36568892921692659120990516883, −5.67618882811943976474042210756, −4.57254999025850795464794371669, −4.01949662092914436300142946813, −3.19448320551249347297506611555, −1.61262689078604032373439999062, 0, 1.61262689078604032373439999062, 3.19448320551249347297506611555, 4.01949662092914436300142946813, 4.57254999025850795464794371669, 5.67618882811943976474042210756, 6.36568892921692659120990516883, 6.72652405724055332287572756453, 8.285098339094794154450601857094, 8.753735690020116685091553768623

Graph of the ZZ-function along the critical line