↑

Properties

Label 2-3822-1.1-c1-0-28
Degree 22
Conductor 38223822
Sign 11
Analytic cond. 30.518830.5188
Root an. cond. 5.524385.52438
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-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 11-s − 12-s + 13-s − 15-s + 16-s + 6·17-s + 18-s + 4·19-s + 20-s − 22-s − 6·23-s − 24-s − 4·25-s + 26-s − 27-s + 3·29-s − 30-s + 11·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.213·22-s − 1.25·23-s − 0.204·24-s − 4/5·25-s + 0.196·26-s − 0.192·27-s + 0.557·29-s − 0.182·30-s + 1.97·31-s + 0.176·32-s + ⋯

Functional equation

Λ(s)=(3822s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3822s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38223822    =    2⋅3⋅72⋅132 \cdot 3 \cdot 7^{2} \cdot 13
Sign: 11
Analytic conductor: 30.518830.5188
Root analytic conductor: 5.524385.52438
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3822, ( :1/2), 1)(2,\ 3822,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.8986508502.898650850
L(12)L(\frac12) ≈\approx 2.8986508502.898650850
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−T 1 - T
3 1+T 1 + T
7 1 1
13 1−T 1 - T
good5 1−T+pT2 1 - T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1−11T+pT2 1 - 11 T + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1+12T+pT2 1 + 12 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+5T+pT2 1 + 5 T + p T^{2}
59 1−5T+pT2 1 - 5 T + p T^{2}
61 1+12T+pT2 1 + 12 T + p T^{2}
67 1−16T+pT2 1 - 16 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1−7T+pT2 1 - 7 T + p T^{2}
83 1−17T+pT2 1 - 17 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1+13T+pT2 1 + 13 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

−8.079464557742516455257811209921, −7.88726865992802290068560784929, −6.67977304881098183980770883507, −6.20968245118849588213967385130, −5.41538490462143317344710809691, −4.94809026004257678850587851082, −3.88685634038772924603182367397, −3.12860410237719280517625053481, −2.03413591199556524513892954961, −0.950220163938610621111634211423, 0.950220163938610621111634211423, 2.03413591199556524513892954961, 3.12860410237719280517625053481, 3.88685634038772924603182367397, 4.94809026004257678850587851082, 5.41538490462143317344710809691, 6.20968245118849588213967385130, 6.67977304881098183980770883507, 7.88726865992802290068560784929, 8.079464557742516455257811209921

Graph of the ZZ-function along the critical line