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Properties

Label 2-4140-1.1-c1-0-21
Degree 22
Conductor 41404140
Sign 11
Analytic cond. 33.058033.0580
Root an. cond. 5.749615.74961
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 5.21·7-s + 2.67·11-s + 0.186·13-s − 1.99·17-s − 0.675·19-s + 23-s + 25-s + 5.02·29-s + 7.52·31-s + 5.21·35-s + 4.34·37-s − 6.60·41-s − 2.84·43-s − 13.5·47-s + 20.1·49-s + 12.9·53-s + 2.67·55-s − 6.74·59-s + 8.65·61-s + 0.186·65-s − 12.1·67-s + 0.181·71-s + 3.81·73-s + 13.9·77-s − 10.4·79-s + 16.8·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.96·7-s + 0.806·11-s + 0.0516·13-s − 0.482·17-s − 0.154·19-s + 0.208·23-s + 0.200·25-s + 0.933·29-s + 1.35·31-s + 0.880·35-s + 0.715·37-s − 1.03·41-s − 0.433·43-s − 1.98·47-s + 2.87·49-s + 1.77·53-s + 0.360·55-s − 0.878·59-s + 1.10·61-s + 0.0230·65-s − 1.48·67-s + 0.0215·71-s + 0.446·73-s + 1.58·77-s − 1.18·79-s + 1.84·83-s + ⋯

Functional equation

Λ(s)=(4140s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4140s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 41404140    =    22⋅32⋅5⋅232^{2} \cdot 3^{2} \cdot 5 \cdot 23
Sign: 11
Analytic conductor: 33.058033.0580
Root analytic conductor: 5.749615.74961
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4140, ( :1/2), 1)(2,\ 4140,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.0018544183.001854418
L(12)L(\frac12) ≈\approx 3.0018544183.001854418
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 1
5 1−T 1 - T
23 1−T 1 - T
good7 1−5.21T+7T2 1 - 5.21T + 7T^{2}
11 1−2.67T+11T2 1 - 2.67T + 11T^{2}
13 1−0.186T+13T2 1 - 0.186T + 13T^{2}
17 1+1.99T+17T2 1 + 1.99T + 17T^{2}
19 1+0.675T+19T2 1 + 0.675T + 19T^{2}
29 1−5.02T+29T2 1 - 5.02T + 29T^{2}
31 1−7.52T+31T2 1 - 7.52T + 31T^{2}
37 1−4.34T+37T2 1 - 4.34T + 37T^{2}
41 1+6.60T+41T2 1 + 6.60T + 41T^{2}
43 1+2.84T+43T2 1 + 2.84T + 43T^{2}
47 1+13.5T+47T2 1 + 13.5T + 47T^{2}
53 1−12.9T+53T2 1 - 12.9T + 53T^{2}
59 1+6.74T+59T2 1 + 6.74T + 59T^{2}
61 1−8.65T+61T2 1 - 8.65T + 61T^{2}
67 1+12.1T+67T2 1 + 12.1T + 67T^{2}
71 1−0.181T+71T2 1 - 0.181T + 71T^{2}
73 1−3.81T+73T2 1 - 3.81T + 73T^{2}
79 1+10.4T+79T2 1 + 10.4T + 79T^{2}
83 1−16.8T+83T2 1 - 16.8T + 83T^{2}
89 1+12.4T+89T2 1 + 12.4T + 89T^{2}
97 1+4.92T+97T2 1 + 4.92T + 97T^{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

−8.442243286716418662687894475482, −7.85713167537143458037708120167, −6.89513225633032120992615731138, −6.28429044370115429211532424112, −5.27695063021709764716262376993, −4.70899864484118438677909233458, −4.07000330528055237724083539442, −2.77086373348407089923421084432, −1.81439403720821503003579893097, −1.10546118893969112856178848140, 1.10546118893969112856178848140, 1.81439403720821503003579893097, 2.77086373348407089923421084432, 4.07000330528055237724083539442, 4.70899864484118438677909233458, 5.27695063021709764716262376993, 6.28429044370115429211532424112, 6.89513225633032120992615731138, 7.85713167537143458037708120167, 8.442243286716418662687894475482

Graph of the ZZ-function along the critical line