↑

Properties

Label 2-4205-1.1-c1-0-239
Degree 22
Conductor 42054205
Sign −1-1
Analytic cond. 33.577033.5770
Root an. cond. 5.794575.79457
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.34·2-s + 1.23·3-s − 0.203·4-s + 5-s + 1.66·6-s − 3.86·7-s − 2.95·8-s − 1.46·9-s + 1.34·10-s + 5.03·11-s − 0.252·12-s + 3.21·13-s − 5.17·14-s + 1.23·15-s − 3.55·16-s − 5.75·17-s − 1.96·18-s + 4.82·19-s − 0.203·20-s − 4.78·21-s + 6.74·22-s − 3.49·23-s − 3.65·24-s + 25-s + 4.30·26-s − 5.53·27-s + 0.787·28-s + ⋯
L(s)  = 1  + 0.947·2-s + 0.715·3-s − 0.101·4-s + 0.447·5-s + 0.677·6-s − 1.46·7-s − 1.04·8-s − 0.488·9-s + 0.423·10-s + 1.51·11-s − 0.0729·12-s + 0.891·13-s − 1.38·14-s + 0.319·15-s − 0.887·16-s − 1.39·17-s − 0.462·18-s + 1.10·19-s − 0.0455·20-s − 1.04·21-s + 1.43·22-s − 0.728·23-s − 0.747·24-s + 0.200·25-s + 0.845·26-s − 1.06·27-s + 0.148·28-s + ⋯

Functional equation

Λ(s)=(4205s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4205s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 42054205    =    5⋅2925 \cdot 29^{2}
Sign: −1-1
Analytic conductor: 33.577033.5770
Root analytic conductor: 5.794575.79457
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4205, ( :1/2), −1)(2,\ 4205,\ (\ :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)
bad5 1−T 1 - T
29 1 1
good2 1−1.34T+2T2 1 - 1.34T + 2T^{2}
3 1−1.23T+3T2 1 - 1.23T + 3T^{2}
7 1+3.86T+7T2 1 + 3.86T + 7T^{2}
11 1−5.03T+11T2 1 - 5.03T + 11T^{2}
13 1−3.21T+13T2 1 - 3.21T + 13T^{2}
17 1+5.75T+17T2 1 + 5.75T + 17T^{2}
19 1−4.82T+19T2 1 - 4.82T + 19T^{2}
23 1+3.49T+23T2 1 + 3.49T + 23T^{2}
31 1+3.10T+31T2 1 + 3.10T + 31T^{2}
37 1−2.71T+37T2 1 - 2.71T + 37T^{2}
41 1+12.5T+41T2 1 + 12.5T + 41T^{2}
43 1+8.64T+43T2 1 + 8.64T + 43T^{2}
47 1−0.222T+47T2 1 - 0.222T + 47T^{2}
53 1+6.68T+53T2 1 + 6.68T + 53T^{2}
59 1+4.63T+59T2 1 + 4.63T + 59T^{2}
61 1+9.46T+61T2 1 + 9.46T + 61T^{2}
67 1+12.9T+67T2 1 + 12.9T + 67T^{2}
71 1−7.50T+71T2 1 - 7.50T + 71T^{2}
73 1−3.53T+73T2 1 - 3.53T + 73T^{2}
79 1+4.45T+79T2 1 + 4.45T + 79T^{2}
83 1−8.43T+83T2 1 - 8.43T + 83T^{2}
89 1−9.68T+89T2 1 - 9.68T + 89T^{2}
97 1+15.6T+97T2 1 + 15.6T + 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

−8.275171829199918371665740362497, −6.95993713598510300003836624216, −6.30294976828677517595916769345, −6.03040635807666292695384059373, −4.98034313660634558272452439237, −3.95693038061179539488680551973, −3.45515359303753619377049891609, −2.91698449877499855988588740568, −1.69925728561776504397360537243, 0, 1.69925728561776504397360537243, 2.91698449877499855988588740568, 3.45515359303753619377049891609, 3.95693038061179539488680551973, 4.98034313660634558272452439237, 6.03040635807666292695384059373, 6.30294976828677517595916769345, 6.95993713598510300003836624216, 8.275171829199918371665740362497

Graph of the ZZ-function along the critical line