↑

Properties

Label 2-1872-1.1-c3-0-31
Degree 22
Conductor 18721872
Sign 11
Analytic cond. 110.451110.451
Root an. cond. 10.509510.5095
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 7.63·5-s − 5.63·7-s + 34.5·11-s + 13·13-s − 2·17-s + 88.1·19-s − 64·23-s − 66.7·25-s − 23.7·29-s + 284.·31-s − 42.9·35-s + 115.·37-s − 1.41·41-s + 337.·43-s − 198.·47-s − 311.·49-s − 59.0·53-s + 263.·55-s − 188.·59-s + 336.·61-s + 99.1·65-s + 531.·67-s − 510.·71-s − 164.·73-s − 194.·77-s + 29.3·79-s − 117.·83-s + ⋯
L(s)  = 1  + 0.682·5-s − 0.303·7-s + 0.946·11-s + 0.277·13-s − 0.0285·17-s + 1.06·19-s − 0.580·23-s − 0.534·25-s − 0.152·29-s + 1.64·31-s − 0.207·35-s + 0.512·37-s − 0.00537·41-s + 1.19·43-s − 0.615·47-s − 0.907·49-s − 0.153·53-s + 0.645·55-s − 0.415·59-s + 0.707·61-s + 0.189·65-s + 0.968·67-s − 0.852·71-s − 0.263·73-s − 0.287·77-s + 0.0417·79-s − 0.155·83-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: 11
Analytic conductor: 110.451110.451
Root analytic conductor: 10.509510.5095
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1872, ( :3/2), 1)(2,\ 1872,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.7753993492.775399349
L(12)L(\frac12) ≈\approx 2.7753993492.775399349
L(52)L(\frac{5}{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 1
13 1−13T 1 - 13T
good5 1−7.63T+125T2 1 - 7.63T + 125T^{2}
7 1+5.63T+343T2 1 + 5.63T + 343T^{2}
11 1−34.5T+1.33e3T2 1 - 34.5T + 1.33e3T^{2}
17 1+2T+4.91e3T2 1 + 2T + 4.91e3T^{2}
19 1−88.1T+6.85e3T2 1 - 88.1T + 6.85e3T^{2}
23 1+64T+1.21e4T2 1 + 64T + 1.21e4T^{2}
29 1+23.7T+2.43e4T2 1 + 23.7T + 2.43e4T^{2}
31 1−284.T+2.97e4T2 1 - 284.T + 2.97e4T^{2}
37 1−115.T+5.06e4T2 1 - 115.T + 5.06e4T^{2}
41 1+1.41T+6.89e4T2 1 + 1.41T + 6.89e4T^{2}
43 1−337.T+7.95e4T2 1 - 337.T + 7.95e4T^{2}
47 1+198.T+1.03e5T2 1 + 198.T + 1.03e5T^{2}
53 1+59.0T+1.48e5T2 1 + 59.0T + 1.48e5T^{2}
59 1+188.T+2.05e5T2 1 + 188.T + 2.05e5T^{2}
61 1−336.T+2.26e5T2 1 - 336.T + 2.26e5T^{2}
67 1−531.T+3.00e5T2 1 - 531.T + 3.00e5T^{2}
71 1+510.T+3.57e5T2 1 + 510.T + 3.57e5T^{2}
73 1+164.T+3.89e5T2 1 + 164.T + 3.89e5T^{2}
79 1−29.3T+4.93e5T2 1 - 29.3T + 4.93e5T^{2}
83 1+117.T+5.71e5T2 1 + 117.T + 5.71e5T^{2}
89 1+508.T+7.04e5T2 1 + 508.T + 7.04e5T^{2}
97 1+1.02e3T+9.12e5T2 1 + 1.02e3T + 9.12e5T^{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

−8.989396734980124287727476614368, −8.127395019059551059899789966059, −7.25327983329754764604097199546, −6.30083719129566073539994324028, −5.89977264098585630320527784739, −4.79809035288805765310844686513, −3.86449315363557137551774524292, −2.91500596122087408259929638650, −1.80052841167742736213425197912, −0.807208050204740756171871700784, 0.807208050204740756171871700784, 1.80052841167742736213425197912, 2.91500596122087408259929638650, 3.86449315363557137551774524292, 4.79809035288805765310844686513, 5.89977264098585630320527784739, 6.30083719129566073539994324028, 7.25327983329754764604097199546, 8.127395019059551059899789966059, 8.989396734980124287727476614368

Graph of the ZZ-function along the critical line