↑

Properties

Label 2-9280-1.1-c1-0-123
Degree 22
Conductor 92809280
Sign 11
Analytic cond. 74.101174.1011
Root an. cond. 8.608208.60820
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.92·3-s + 5-s + 3.41·7-s + 0.717·9-s + 4.88·11-s − 1.11·13-s + 1.92·15-s − 3.64·17-s + 1.58·19-s + 6.58·21-s − 8.13·23-s + 25-s − 4.40·27-s + 29-s + 6.71·31-s + 9.42·33-s + 3.41·35-s + 2.06·37-s − 2.15·39-s + 5.19·41-s + 10.1·43-s + 0.717·45-s + 8.48·47-s + 4.68·49-s − 7.01·51-s − 10.8·53-s + 4.88·55-s + ⋯
L(s)  = 1  + 1.11·3-s + 0.447·5-s + 1.29·7-s + 0.239·9-s + 1.47·11-s − 0.309·13-s + 0.497·15-s − 0.882·17-s + 0.364·19-s + 1.43·21-s − 1.69·23-s + 0.200·25-s − 0.846·27-s + 0.185·29-s + 1.20·31-s + 1.64·33-s + 0.577·35-s + 0.339·37-s − 0.344·39-s + 0.810·41-s + 1.55·43-s + 0.106·45-s + 1.23·47-s + 0.668·49-s − 0.982·51-s − 1.49·53-s + 0.659·55-s + ⋯

Functional equation

Λ(s)=(9280s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9280s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 92809280    =    26⋅5⋅292^{6} \cdot 5 \cdot 29
Sign: 11
Analytic conductor: 74.101174.1011
Root analytic conductor: 8.608208.60820
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9280, ( :1/2), 1)(2,\ 9280,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.4655672934.465567293
L(12)L(\frac12) ≈\approx 4.4655672934.465567293
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 1
5 1−T 1 - T
29 1−T 1 - T
good3 1−1.92T+3T2 1 - 1.92T + 3T^{2}
7 1−3.41T+7T2 1 - 3.41T + 7T^{2}
11 1−4.88T+11T2 1 - 4.88T + 11T^{2}
13 1+1.11T+13T2 1 + 1.11T + 13T^{2}
17 1+3.64T+17T2 1 + 3.64T + 17T^{2}
19 1−1.58T+19T2 1 - 1.58T + 19T^{2}
23 1+8.13T+23T2 1 + 8.13T + 23T^{2}
31 1−6.71T+31T2 1 - 6.71T + 31T^{2}
37 1−2.06T+37T2 1 - 2.06T + 37T^{2}
41 1−5.19T+41T2 1 - 5.19T + 41T^{2}
43 1−10.1T+43T2 1 - 10.1T + 43T^{2}
47 1−8.48T+47T2 1 - 8.48T + 47T^{2}
53 1+10.8T+53T2 1 + 10.8T + 53T^{2}
59 1−10.9T+59T2 1 - 10.9T + 59T^{2}
61 1+14.2T+61T2 1 + 14.2T + 61T^{2}
67 1−7.50T+67T2 1 - 7.50T + 67T^{2}
71 1−9.45T+71T2 1 - 9.45T + 71T^{2}
73 1+11.5T+73T2 1 + 11.5T + 73T^{2}
79 1+4.83T+79T2 1 + 4.83T + 79T^{2}
83 1−11.0T+83T2 1 - 11.0T + 83T^{2}
89 1+10.6T+89T2 1 + 10.6T + 89T^{2}
97 1−18.2T+97T2 1 - 18.2T + 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

−7.79532318879636238237173404169, −7.27294972054433684934795081504, −6.24484651471766296152635008020, −5.83986401640519521977036952736, −4.62026372314784534183114862744, −4.29789819656543074556240088579, −3.43857626760359377026770431658, −2.37659138727350784648141047916, −1.98346944974395038378352768445, −1.01141507522903470635733248117, 1.01141507522903470635733248117, 1.98346944974395038378352768445, 2.37659138727350784648141047916, 3.43857626760359377026770431658, 4.29789819656543074556240088579, 4.62026372314784534183114862744, 5.83986401640519521977036952736, 6.24484651471766296152635008020, 7.27294972054433684934795081504, 7.79532318879636238237173404169

Graph of the ZZ-function along the critical line