↑

Properties

Label 2-3192-1.1-c1-0-55
Degree 22
Conductor 31923192
Sign −1-1
Analytic cond. 25.488225.4882
Root an. cond. 5.048585.04858
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 0.732·5-s + 7-s + 9-s − 4·11-s − 1.46·13-s + 0.732·15-s − 3.26·17-s + 19-s + 21-s − 6.92·23-s − 4.46·25-s + 27-s − 3.26·29-s − 2·31-s − 4·33-s + 0.732·35-s − 4.92·37-s − 1.46·39-s − 8.92·41-s + 4.92·43-s + 0.732·45-s + 0.196·47-s + 49-s − 3.26·51-s + 7.66·53-s − 2.92·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.327·5-s + 0.377·7-s + 0.333·9-s − 1.20·11-s − 0.406·13-s + 0.189·15-s − 0.792·17-s + 0.229·19-s + 0.218·21-s − 1.44·23-s − 0.892·25-s + 0.192·27-s − 0.606·29-s − 0.359·31-s − 0.696·33-s + 0.123·35-s − 0.810·37-s − 0.234·39-s − 1.39·41-s + 0.751·43-s + 0.109·45-s + 0.0286·47-s + 0.142·49-s − 0.457·51-s + 1.05·53-s − 0.394·55-s + ⋯

Functional equation

Λ(s)=(3192s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3192s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31923192    =    23⋅3⋅7⋅192^{3} \cdot 3 \cdot 7 \cdot 19
Sign: −1-1
Analytic conductor: 25.488225.4882
Root analytic conductor: 5.048585.04858
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3192, ( :1/2), −1)(2,\ 3192,\ (\ :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)
bad2 1 1
3 1−T 1 - T
7 1−T 1 - T
19 1−T 1 - T
good5 1−0.732T+5T2 1 - 0.732T + 5T^{2}
11 1+4T+11T2 1 + 4T + 11T^{2}
13 1+1.46T+13T2 1 + 1.46T + 13T^{2}
17 1+3.26T+17T2 1 + 3.26T + 17T^{2}
23 1+6.92T+23T2 1 + 6.92T + 23T^{2}
29 1+3.26T+29T2 1 + 3.26T + 29T^{2}
31 1+2T+31T2 1 + 2T + 31T^{2}
37 1+4.92T+37T2 1 + 4.92T + 37T^{2}
41 1+8.92T+41T2 1 + 8.92T + 41T^{2}
43 1−4.92T+43T2 1 - 4.92T + 43T^{2}
47 1−0.196T+47T2 1 - 0.196T + 47T^{2}
53 1−7.66T+53T2 1 - 7.66T + 53T^{2}
59 1+10.9T+59T2 1 + 10.9T + 59T^{2}
61 1−0.928T+61T2 1 - 0.928T + 61T^{2}
67 1−5.46T+67T2 1 - 5.46T + 67T^{2}
71 1−4.19T+71T2 1 - 4.19T + 71T^{2}
73 1+10.3T+73T2 1 + 10.3T + 73T^{2}
79 1−2.92T+79T2 1 - 2.92T + 79T^{2}
83 1−8.19T+83T2 1 - 8.19T + 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1−11.8T+97T2 1 - 11.8T + 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

−8.215080694543461831295586226946, −7.69077403006010244619603218209, −6.96597955757863350195984661075, −5.91836790843344527318839669470, −5.24732949074927538314814056862, −4.38183196384062589024908734169, −3.48395451475226006682558209041, −2.38738853561610578284772266510, −1.83437597084645015196398689254, 0, 1.83437597084645015196398689254, 2.38738853561610578284772266510, 3.48395451475226006682558209041, 4.38183196384062589024908734169, 5.24732949074927538314814056862, 5.91836790843344527318839669470, 6.96597955757863350195984661075, 7.69077403006010244619603218209, 8.215080694543461831295586226946

Graph of the ZZ-function along the critical line