↑

Properties

Label 2-41971-1.1-c1-0-0
Degree 22
Conductor 4197141971
Sign 11
Analytic cond. 335.140335.140
Root an. cond. 18.306818.3068
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·4-s − 3·5-s − 7-s + 9-s − 3·11-s + 4·12-s + 4·13-s + 6·15-s + 4·16-s − 3·17-s − 19-s + 6·20-s + 2·21-s + 4·25-s + 4·27-s + 2·28-s − 6·29-s + 4·31-s + 6·33-s + 3·35-s − 2·36-s + 2·37-s − 8·39-s + 6·41-s + 43-s + 6·44-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 1.15·12-s + 1.10·13-s + 1.54·15-s + 16-s − 0.727·17-s − 0.229·19-s + 1.34·20-s + 0.436·21-s + 4/5·25-s + 0.769·27-s + 0.377·28-s − 1.11·29-s + 0.718·31-s + 1.04·33-s + 0.507·35-s − 1/3·36-s + 0.328·37-s − 1.28·39-s + 0.937·41-s + 0.152·43-s + 0.904·44-s + ⋯

Functional equation

Λ(s)=(41971s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 41971 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(41971s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 41971 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4197141971    =    19⋅47219 \cdot 47^{2}
Sign: 11
Analytic conductor: 335.140335.140
Root analytic conductor: 18.306818.3068
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 41971, ( :1/2), 1)(2,\ 41971,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.60199829660.6019982966
L(12)L(\frac12) ≈\approx 0.60199829660.6019982966
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)
bad19 1+T 1 + T
47 1 1
good2 1+pT2 1 + p T^{2}
3 1+2T+pT2 1 + 2 T + p T^{2}
5 1+3T+pT2 1 + 3 T + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1+T+pT2 1 + T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1−7T+pT2 1 - 7 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1−8T+pT2 1 - 8 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

−14.82062993720606, −14.17554838271630, −13.50924567493881, −13.03471210825680, −12.72711230000074, −12.07151758943378, −11.59054147429658, −11.08182005207214, −10.69725351046066, −10.20465328287405, −9.376933256368812, −8.872329961180960, −8.300762533956756, −7.840381302542135, −7.305031007172905, −6.411634217398548, −6.066246078508152, −5.369590393183792, −4.805546356798797, −4.314432567977842, −3.679847158564805, −3.208837694698318, −2.132348298031530, −0.7432226443089781, −0.4915439028913777, 0.4915439028913777, 0.7432226443089781, 2.132348298031530, 3.208837694698318, 3.679847158564805, 4.314432567977842, 4.805546356798797, 5.369590393183792, 6.066246078508152, 6.411634217398548, 7.305031007172905, 7.840381302542135, 8.300762533956756, 8.872329961180960, 9.376933256368812, 10.20465328287405, 10.69725351046066, 11.08182005207214, 11.59054147429658, 12.07151758943378, 12.72711230000074, 13.03471210825680, 13.50924567493881, 14.17554838271630, 14.82062993720606

Graph of the ZZ-function along the critical line