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Properties

Label 2-960-1.1-c1-0-4
Degree 22
Conductor 960960
Sign 11
Analytic cond. 7.665637.66563
Root an. cond. 2.768682.76868
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 − 4·7-s + 9-s + 6·13-s + 15-s − 2·17-s + 4·19-s − 4·21-s + 8·23-s + 25-s + 27-s + 6·29-s − 4·35-s + 6·37-s + 6·39-s + 10·41-s − 4·43-s + 45-s − 8·47-s + 9·49-s − 2·51-s − 10·53-s + 4·57-s − 6·61-s − 4·63-s + 6·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.66·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s − 0.872·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.676·35-s + 0.986·37-s + 0.960·39-s + 1.56·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s + 9/7·49-s − 0.280·51-s − 1.37·53-s + 0.529·57-s − 0.768·61-s − 0.503·63-s + 0.744·65-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: 11
Analytic conductor: 7.665637.66563
Root analytic conductor: 2.768682.76868
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 960, ( :1/2), 1)(2,\ 960,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9775352501.977535250
L(12)L(\frac12) ≈\approx 1.9775352501.977535250
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+4T+pT2 1 + 4 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1−4T+pT2 1 - 4 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+pT2 1 + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+14T+pT2 1 + 14 T + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−2T+pT2 1 - 2 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

−9.843986395238725085410957067541, −9.192233856919801710529869731712, −8.620274137521206455931441584726, −7.45166089807995498253197140252, −6.49211773175788373093258725474, −6.00871420537581383609527234257, −4.63883887090438359185158186412, −3.39015414339768891989414504202, −2.85965731186513405981248890470, −1.16849564057756810686392475492, 1.16849564057756810686392475492, 2.85965731186513405981248890470, 3.39015414339768891989414504202, 4.63883887090438359185158186412, 6.00871420537581383609527234257, 6.49211773175788373093258725474, 7.45166089807995498253197140252, 8.620274137521206455931441584726, 9.192233856919801710529869731712, 9.843986395238725085410957067541

Graph of the ZZ-function along the critical line