↑

Properties

Label 2-675-1.1-c3-0-3
Degree 22
Conductor 675675
Sign 11
Analytic cond. 39.826239.8262
Root an. cond. 6.310806.31080
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  + 0.561·2-s − 7.68·4-s − 29.8·7-s − 8.80·8-s − 56.0·11-s + 20.4·13-s − 16.7·14-s + 56.5·16-s − 18.5·17-s − 70.7·19-s − 31.4·22-s − 111.·23-s + 11.4·26-s + 229.·28-s − 72.7·29-s + 106.·31-s + 102.·32-s − 10.4·34-s + 100.·37-s − 39.7·38-s + 307.·41-s − 479.·43-s + 430.·44-s − 62.8·46-s + 472.·47-s + 547.·49-s − 156.·52-s + ⋯
L(s)  = 1  + 0.198·2-s − 0.960·4-s − 1.61·7-s − 0.389·8-s − 1.53·11-s + 0.435·13-s − 0.319·14-s + 0.883·16-s − 0.264·17-s − 0.854·19-s − 0.304·22-s − 1.01·23-s + 0.0865·26-s + 1.54·28-s − 0.465·29-s + 0.615·31-s + 0.564·32-s − 0.0525·34-s + 0.446·37-s − 0.169·38-s + 1.17·41-s − 1.69·43-s + 1.47·44-s − 0.201·46-s + 1.46·47-s + 1.59·49-s − 0.418·52-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: 11
Analytic conductor: 39.826239.8262
Root analytic conductor: 6.310806.31080
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 675, ( :3/2), 1)(2,\ 675,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.55331690390.5533169039
L(12)L(\frac12) ≈\approx 0.55331690390.5533169039
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)
bad3 1 1
5 1 1
good2 1−0.561T+8T2 1 - 0.561T + 8T^{2}
7 1+29.8T+343T2 1 + 29.8T + 343T^{2}
11 1+56.0T+1.33e3T2 1 + 56.0T + 1.33e3T^{2}
13 1−20.4T+2.19e3T2 1 - 20.4T + 2.19e3T^{2}
17 1+18.5T+4.91e3T2 1 + 18.5T + 4.91e3T^{2}
19 1+70.7T+6.85e3T2 1 + 70.7T + 6.85e3T^{2}
23 1+111.T+1.21e4T2 1 + 111.T + 1.21e4T^{2}
29 1+72.7T+2.43e4T2 1 + 72.7T + 2.43e4T^{2}
31 1−106.T+2.97e4T2 1 - 106.T + 2.97e4T^{2}
37 1−100.T+5.06e4T2 1 - 100.T + 5.06e4T^{2}
41 1−307.T+6.89e4T2 1 - 307.T + 6.89e4T^{2}
43 1+479.T+7.95e4T2 1 + 479.T + 7.95e4T^{2}
47 1−472.T+1.03e5T2 1 - 472.T + 1.03e5T^{2}
53 1−583.T+1.48e5T2 1 - 583.T + 1.48e5T^{2}
59 1+429.T+2.05e5T2 1 + 429.T + 2.05e5T^{2}
61 1+443.T+2.26e5T2 1 + 443.T + 2.26e5T^{2}
67 1−465.T+3.00e5T2 1 - 465.T + 3.00e5T^{2}
71 1+1.05e3T+3.57e5T2 1 + 1.05e3T + 3.57e5T^{2}
73 1+234.T+3.89e5T2 1 + 234.T + 3.89e5T^{2}
79 1−275.T+4.93e5T2 1 - 275.T + 4.93e5T^{2}
83 1−1.18e3T+5.71e5T2 1 - 1.18e3T + 5.71e5T^{2}
89 1+309.T+7.04e5T2 1 + 309.T + 7.04e5T^{2}
97 1−637.T+9.12e5T2 1 - 637.T + 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

−10.12133302774640062505865409940, −9.277610260670559263246374486900, −8.477039130836817883123303564916, −7.55576800353488664800154960802, −6.28980871927789693624363328836, −5.67427364799795083103387043053, −4.48252339822949814848647305595, −3.54279313649687255228512272630, −2.53253139979344015134078415680, −0.39703553741338774913318063482, 0.39703553741338774913318063482, 2.53253139979344015134078415680, 3.54279313649687255228512272630, 4.48252339822949814848647305595, 5.67427364799795083103387043053, 6.28980871927789693624363328836, 7.55576800353488664800154960802, 8.477039130836817883123303564916, 9.277610260670559263246374486900, 10.12133302774640062505865409940

Graph of the ZZ-function along the critical line