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Properties

Label 2-960-1.1-c1-0-5
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 + 9-s + 4·11-s + 2·13-s − 15-s + 2·17-s − 4·19-s + 25-s + 27-s + 2·29-s + 4·33-s + 10·37-s + 2·39-s + 10·41-s − 4·43-s − 45-s + 8·47-s − 7·49-s + 2·51-s + 10·53-s − 4·55-s − 4·57-s + 4·59-s + 2·61-s − 2·65-s − 12·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.696·33-s + 1.64·37-s + 0.320·39-s + 1.56·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s − 49-s + 0.280·51-s + 1.37·53-s − 0.539·55-s − 0.529·57-s + 0.520·59-s + 0.256·61-s − 0.248·65-s − 1.46·67-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 1.9807518171.980751817
L(12)L(\frac12) ≈\approx 1.9807518171.980751817
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 1−2T+pT2 1 - 2 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+pT2 1 + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−10T+pT2 1 - 10 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 1−4T+pT2 1 - 4 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+6T+pT2 1 + 6 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

−9.907104671360377962786893632766, −9.080341213166513123570518780286, −8.438902886313495638742074033009, −7.60125180356652570547951985370, −6.67875901406960725991694880905, −5.85130665991850657402735978605, −4.38871739502786245271092399731, −3.82349173075266898382654912582, −2.63252540175087701435373844808, −1.18815390030287753587945796282, 1.18815390030287753587945796282, 2.63252540175087701435373844808, 3.82349173075266898382654912582, 4.38871739502786245271092399731, 5.85130665991850657402735978605, 6.67875901406960725991694880905, 7.60125180356652570547951985370, 8.438902886313495638742074033009, 9.080341213166513123570518780286, 9.907104671360377962786893632766

Graph of the ZZ-function along the critical line