↑

Properties

Label 2-110670-1.1-c1-0-18
Degree 22
Conductor 110670110670
Sign 11
Analytic cond. 883.704883.704
Root an. cond. 29.727129.7271
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 + 7-s − 8-s + 9-s − 10-s + 12-s − 13-s − 14-s + 15-s + 16-s + 17-s − 18-s + 5·19-s + 20-s + 21-s − 3·23-s − 24-s + 25-s + 26-s + 27-s + 28-s − 3·29-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.14·19-s + 0.223·20-s + 0.218·21-s − 0.625·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s − 0.557·29-s − 0.182·30-s + ⋯

Functional equation

Λ(s)=(110670s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(110670s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110670110670    =    2⋅3⋅5⋅7⋅17⋅312 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 31
Sign: 11
Analytic conductor: 883.704883.704
Root analytic conductor: 29.727129.7271
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 110670, ( :1/2), 1)(2,\ 110670,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.1823318393.182331839
L(12)L(\frac12) ≈\approx 3.1823318393.182331839
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
5 1−T 1 - T
7 1−T 1 - T
17 1−T 1 - T
31 1−T 1 - T
good11 1+pT2 1 + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
19 1−5T+pT2 1 - 5 T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
37 1−11T+pT2 1 - 11 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1+3T+pT2 1 + 3 T + p T^{2}
59 1−12T+pT2 1 - 12 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+12T+pT2 1 + 12 T + p T^{2}
73 1+T+pT2 1 + T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1−6T+pT2 1 - 6 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

−13.67638217717226, −13.19355684126266, −12.73199190837579, −12.05538496336393, −11.72447540997303, −11.13121189143117, −10.61105635790118, −10.13267910354687, −9.549663832590038, −9.326221184852871, −8.829194907221277, −8.138466929673498, −7.604179739423918, −7.526891896093217, −6.745536822738579, −6.076009259524598, −5.644418270185105, −5.080996940800354, −4.233477151332721, −3.879154485528115, −2.867943356012189, −2.635273443994216, −1.912127286089923, −1.225164838238428, −0.6404644260064002, 0.6404644260064002, 1.225164838238428, 1.912127286089923, 2.635273443994216, 2.867943356012189, 3.879154485528115, 4.233477151332721, 5.080996940800354, 5.644418270185105, 6.076009259524598, 6.745536822738579, 7.526891896093217, 7.604179739423918, 8.138466929673498, 8.829194907221277, 9.326221184852871, 9.549663832590038, 10.13267910354687, 10.61105635790118, 11.13121189143117, 11.72447540997303, 12.05538496336393, 12.73199190837579, 13.19355684126266, 13.67638217717226

Graph of the ZZ-function along the critical line