↑

Properties

Label 2-120-1.1-c3-0-4
Degree 22
Conductor 120120
Sign −1-1
Analytic cond. 7.080227.08022
Root an. cond. 2.660872.66087
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 5·5-s + 20·7-s + 9·9-s − 56·11-s − 86·13-s + 15·15-s − 106·17-s + 4·19-s − 60·21-s + 136·23-s + 25·25-s − 27·27-s − 206·29-s − 152·31-s + 168·33-s − 100·35-s + 282·37-s + 258·39-s − 246·41-s + 412·43-s − 45·45-s + 40·47-s + 57·49-s + 318·51-s − 126·53-s + 280·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1.07·7-s + 1/3·9-s − 1.53·11-s − 1.83·13-s + 0.258·15-s − 1.51·17-s + 0.0482·19-s − 0.623·21-s + 1.23·23-s + 1/5·25-s − 0.192·27-s − 1.31·29-s − 0.880·31-s + 0.886·33-s − 0.482·35-s + 1.25·37-s + 1.05·39-s − 0.937·41-s + 1.46·43-s − 0.149·45-s + 0.124·47-s + 0.166·49-s + 0.873·51-s − 0.326·53-s + 0.686·55-s + ⋯

Functional equation

Λ(s)=(120s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(120s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: −1-1
Analytic conductor: 7.080227.08022
Root analytic conductor: 2.660872.66087
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 120, ( :3/2), −1)(2,\ 120,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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+pT 1 + p T
5 1+pT 1 + p T
good7 1−20T+p3T2 1 - 20 T + p^{3} T^{2}
11 1+56T+p3T2 1 + 56 T + p^{3} T^{2}
13 1+86T+p3T2 1 + 86 T + p^{3} T^{2}
17 1+106T+p3T2 1 + 106 T + p^{3} T^{2}
19 1−4T+p3T2 1 - 4 T + p^{3} T^{2}
23 1−136T+p3T2 1 - 136 T + p^{3} T^{2}
29 1+206T+p3T2 1 + 206 T + p^{3} T^{2}
31 1+152T+p3T2 1 + 152 T + p^{3} T^{2}
37 1−282T+p3T2 1 - 282 T + p^{3} T^{2}
41 1+6pT+p3T2 1 + 6 p T + p^{3} T^{2}
43 1−412T+p3T2 1 - 412 T + p^{3} T^{2}
47 1−40T+p3T2 1 - 40 T + p^{3} T^{2}
53 1+126T+p3T2 1 + 126 T + p^{3} T^{2}
59 1−56T+p3T2 1 - 56 T + p^{3} T^{2}
61 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
67 1+388T+p3T2 1 + 388 T + p^{3} T^{2}
71 1+672T+p3T2 1 + 672 T + p^{3} T^{2}
73 1−1170T+p3T2 1 - 1170 T + p^{3} T^{2}
79 1−408T+p3T2 1 - 408 T + p^{3} T^{2}
83 1−668T+p3T2 1 - 668 T + p^{3} T^{2}
89 1−66T+p3T2 1 - 66 T + p^{3} T^{2}
97 1+926T+p3T2 1 + 926 T + p^{3} 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

−12.46463514703776210037668278964, −11.26572264432476285563810637861, −10.74754642118653737554343963329, −9.329803339228465296065782693661, −7.88380481382571055360508795733, −7.16023597148262679847623851296, −5.30655493261829368208922248556, −4.59991571390655140899607859969, −2.36344668504755659920796853295, 0, 2.36344668504755659920796853295, 4.59991571390655140899607859969, 5.30655493261829368208922248556, 7.16023597148262679847623851296, 7.88380481382571055360508795733, 9.329803339228465296065782693661, 10.74754642118653737554343963329, 11.26572264432476285563810637861, 12.46463514703776210037668278964

Graph of the ZZ-function along the critical line