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Properties

Label 2-96e2-1.1-c1-0-78
Degree 22
Conductor 92169216
Sign 11
Analytic cond. 73.590173.5901
Root an. cond. 8.578468.57846
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  + 1.41·5-s + 4·7-s + 5.65·11-s − 4.24·13-s + 6·17-s + 5.65·19-s − 8·23-s − 2.99·25-s − 4.24·29-s + 4·31-s + 5.65·35-s − 1.41·37-s + 2·41-s − 5.65·43-s + 8·47-s + 9·49-s + 9.89·53-s + 8.00·55-s − 4.24·61-s − 6·65-s + 11.3·67-s + 10·73-s + 22.6·77-s + 12·79-s + 5.65·83-s + 8.48·85-s − 16·89-s + ⋯
L(s)  = 1  + 0.632·5-s + 1.51·7-s + 1.70·11-s − 1.17·13-s + 1.45·17-s + 1.29·19-s − 1.66·23-s − 0.599·25-s − 0.787·29-s + 0.718·31-s + 0.956·35-s − 0.232·37-s + 0.312·41-s − 0.862·43-s + 1.16·47-s + 1.28·49-s + 1.35·53-s + 1.07·55-s − 0.543·61-s − 0.744·65-s + 1.38·67-s + 1.17·73-s + 2.57·77-s + 1.35·79-s + 0.620·83-s + 0.920·85-s − 1.69·89-s + ⋯

Functional equation

Λ(s)=(9216s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9216s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 92169216    =    210⋅322^{10} \cdot 3^{2}
Sign: 11
Analytic conductor: 73.590173.5901
Root analytic conductor: 8.578468.57846
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9216, ( :1/2), 1)(2,\ 9216,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.4919334163.491933416
L(12)L(\frac12) ≈\approx 3.4919334163.491933416
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
good5 1−1.41T+5T2 1 - 1.41T + 5T^{2}
7 1−4T+7T2 1 - 4T + 7T^{2}
11 1−5.65T+11T2 1 - 5.65T + 11T^{2}
13 1+4.24T+13T2 1 + 4.24T + 13T^{2}
17 1−6T+17T2 1 - 6T + 17T^{2}
19 1−5.65T+19T2 1 - 5.65T + 19T^{2}
23 1+8T+23T2 1 + 8T + 23T^{2}
29 1+4.24T+29T2 1 + 4.24T + 29T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1+1.41T+37T2 1 + 1.41T + 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1+5.65T+43T2 1 + 5.65T + 43T^{2}
47 1−8T+47T2 1 - 8T + 47T^{2}
53 1−9.89T+53T2 1 - 9.89T + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+4.24T+61T2 1 + 4.24T + 61T^{2}
67 1−11.3T+67T2 1 - 11.3T + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−10T+73T2 1 - 10T + 73T^{2}
79 1−12T+79T2 1 - 12T + 79T^{2}
83 1−5.65T+83T2 1 - 5.65T + 83T^{2}
89 1+16T+89T2 1 + 16T + 89T^{2}
97 1−8T+97T2 1 - 8T + 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

−7.73536282952643018202134004908, −7.17896038249826618637825140691, −6.27475982917468646684113643983, −5.51064588113498983081165322552, −5.14928741387013799069560225998, −4.16596500324326316740245778538, −3.62793282621066475013107164267, −2.37921804647973866560352906238, −1.69213857367136046510671702177, −0.988736452349166151317044692196, 0.988736452349166151317044692196, 1.69213857367136046510671702177, 2.37921804647973866560352906238, 3.62793282621066475013107164267, 4.16596500324326316740245778538, 5.14928741387013799069560225998, 5.51064588113498983081165322552, 6.27475982917468646684113643983, 7.17896038249826618637825140691, 7.73536282952643018202134004908

Graph of the ZZ-function along the critical line