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Properties

Label 2-120-1.1-c1-0-1
Degree 22
Conductor 120120
Sign 11
Analytic cond. 0.9582040.958204
Root an. cond. 0.9788790.978879
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 9-s − 4·11-s + 6·13-s + 15-s − 6·17-s − 4·19-s + 25-s + 27-s − 2·29-s − 8·31-s − 4·33-s − 2·37-s + 6·39-s − 6·41-s + 12·43-s + 45-s + 8·47-s − 7·49-s − 6·51-s + 6·53-s − 4·55-s − 4·57-s + 12·59-s + 14·61-s + 6·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s − 0.328·37-s + 0.960·39-s − 0.937·41-s + 1.82·43-s + 0.149·45-s + 1.16·47-s − 49-s − 0.840·51-s + 0.824·53-s − 0.539·55-s − 0.529·57-s + 1.56·59-s + 1.79·61-s + 0.744·65-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 120120    =    23⋅3⋅52^{3} \cdot 3 \cdot 5
Sign: 11
Analytic conductor: 0.9582040.958204
Root analytic conductor: 0.9788790.978879
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 120, ( :1/2), 1)(2,\ 120,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2694942771.269494277
L(12)L(\frac12) ≈\approx 1.2694942771.269494277
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
5 1−T 1 - T
good7 1+pT2 1 + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−12T+pT2 1 - 12 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1−2T+pT2 1 - 2 T + p T^{2}
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   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

−13.23486091060546125236692720333, −13.00221841062232627578538988126, −11.15225413084276142148114637688, −10.45550448318589263597286780431, −9.042960162718976648409639393560, −8.323057177226078640880314002796, −6.89270810769313104844169224865, −5.59585158632682227968586342616, −3.95303834850677407956821293662, −2.25002432004047770108316173263, 2.25002432004047770108316173263, 3.95303834850677407956821293662, 5.59585158632682227968586342616, 6.89270810769313104844169224865, 8.323057177226078640880314002796, 9.042960162718976648409639393560, 10.45550448318589263597286780431, 11.15225413084276142148114637688, 13.00221841062232627578538988126, 13.23486091060546125236692720333

Graph of the ZZ-function along the critical line