↑

Properties

Label 2-2535-1.1-c1-0-12
Degree 22
Conductor 25352535
Sign 11
Analytic cond. 20.242020.2420
Root an. cond. 4.499114.49911
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  − 3-s − 2·4-s − 5-s + 3·7-s + 9-s − 3·11-s + 2·12-s + 15-s + 4·16-s − 3·17-s + 2·20-s − 3·21-s + 3·23-s + 25-s − 27-s − 6·28-s − 6·29-s + 6·31-s + 3·33-s − 3·35-s − 2·36-s − 9·37-s − 3·41-s + 10·43-s + 6·44-s − 45-s − 12·47-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.447·5-s + 1.13·7-s + 1/3·9-s − 0.904·11-s + 0.577·12-s + 0.258·15-s + 16-s − 0.727·17-s + 0.447·20-s − 0.654·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s − 1.13·28-s − 1.11·29-s + 1.07·31-s + 0.522·33-s − 0.507·35-s − 1/3·36-s − 1.47·37-s − 0.468·41-s + 1.52·43-s + 0.904·44-s − 0.149·45-s − 1.75·47-s + ⋯

Functional equation

Λ(s)=(2535s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2535s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25352535    =    3⋅5⋅1323 \cdot 5 \cdot 13^{2}
Sign: 11
Analytic conductor: 20.242020.2420
Root analytic conductor: 4.499114.49911
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2535, ( :1/2), 1)(2,\ 2535,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.88924451230.8892445123
L(12)L(\frac12) ≈\approx 0.88924451230.8892445123
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+T 1 + T
13 1 1
good2 1+pT2 1 + p T^{2}
7 1−3T+pT2 1 - 3 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−6T+pT2 1 - 6 T + p T^{2}
37 1+9T+pT2 1 + 9 T + p T^{2}
41 1+3T+pT2 1 + 3 T + p T^{2}
43 1−10T+pT2 1 - 10 T + p T^{2}
47 1+12T+pT2 1 + 12 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+pT2 1 + p T^{2}
71 1−9T+pT2 1 - 9 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1−T+pT2 1 - T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1−15T+pT2 1 - 15 T + p T^{2}
97 1−9T+pT2 1 - 9 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

−8.765154230348609914473829791484, −8.162800629217765674167583246921, −7.56669985510585965966341183018, −6.61462902511927818987493434872, −5.44114193130814224462783205875, −4.98904539167326410769878999374, −4.36403452129879769416159437543, −3.38037129798030626259220459654, −1.95733443232044255651478454976, −0.61990195475778641136919885683, 0.61990195475778641136919885683, 1.95733443232044255651478454976, 3.38037129798030626259220459654, 4.36403452129879769416159437543, 4.98904539167326410769878999374, 5.44114193130814224462783205875, 6.61462902511927818987493434872, 7.56669985510585965966341183018, 8.162800629217765674167583246921, 8.765154230348609914473829791484

Graph of the ZZ-function along the critical line