↑

Properties

Label 2-462-1.1-c1-0-8
Degree 22
Conductor 462462
Sign −1-1
Analytic cond. 3.689083.68908
Root an. cond. 1.920701.92070
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 4·5-s − 6-s + 7-s + 8-s + 9-s − 4·10-s − 11-s − 12-s − 6·13-s + 14-s + 4·15-s + 16-s − 4·17-s + 18-s − 2·19-s − 4·20-s − 21-s − 22-s − 8·23-s − 24-s + 11·25-s − 6·26-s − 27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.301·11-s − 0.288·12-s − 1.66·13-s + 0.267·14-s + 1.03·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.458·19-s − 0.894·20-s − 0.218·21-s − 0.213·22-s − 1.66·23-s − 0.204·24-s + 11/5·25-s − 1.17·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

Λ(s)=(462s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(462s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 462462    =    2⋅3⋅7⋅112 \cdot 3 \cdot 7 \cdot 11
Sign: −1-1
Analytic conductor: 3.689083.68908
Root analytic conductor: 1.920701.92070
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 462, ( :1/2), −1)(2,\ 462,\ (\ :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−T 1 - T
3 1+T 1 + T
7 1−T 1 - T
11 1+T 1 + T
good5 1+4T+pT2 1 + 4 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−6T+pT2 1 - 6 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1−12T+pT2 1 - 12 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1+pT2 1 + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1+14T+pT2 1 + 14 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1+14T+pT2 1 + 14 T + p T^{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

−10.99555820546878314855420687836, −10.01228279213210772511711884000, −8.515253748126092137293786902708, −7.56853928475057757956492711754, −7.06196128452260537589494409619, −5.68884256231510340821014445655, −4.50182617970418156245344086250, −4.10732223636165202223499092609, −2.48203239562124190952326585155, 0, 2.48203239562124190952326585155, 4.10732223636165202223499092609, 4.50182617970418156245344086250, 5.68884256231510340821014445655, 7.06196128452260537589494409619, 7.56853928475057757956492711754, 8.515253748126092137293786902708, 10.01228279213210772511711884000, 10.99555820546878314855420687836

Graph of the ZZ-function along the critical line