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Properties

Label 2-616-1.1-c1-0-9
Degree 22
Conductor 616616
Sign 11
Analytic cond. 4.918784.91878
Root an. cond. 2.217832.21783
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  + 2.68·3-s + 2.68·5-s + 7-s + 4.18·9-s + 11-s − 2.50·13-s + 7.18·15-s − 6.37·17-s − 3.87·19-s + 2.68·21-s − 4.55·23-s + 2.18·25-s + 3.18·27-s + 3.01·29-s + 5.18·31-s + 2.68·33-s + 2.68·35-s − 6.55·37-s − 6.72·39-s − 4.34·41-s − 1.01·43-s + 11.2·45-s + 0.637·47-s + 49-s − 17.1·51-s − 3.01·53-s + 2.68·55-s + ⋯
L(s)  = 1  + 1.54·3-s + 1.19·5-s + 0.377·7-s + 1.39·9-s + 0.301·11-s − 0.695·13-s + 1.85·15-s − 1.54·17-s − 0.888·19-s + 0.585·21-s − 0.949·23-s + 0.437·25-s + 0.613·27-s + 0.560·29-s + 0.932·31-s + 0.466·33-s + 0.453·35-s − 1.07·37-s − 1.07·39-s − 0.678·41-s − 0.155·43-s + 1.67·45-s + 0.0929·47-s + 0.142·49-s − 2.39·51-s − 0.414·53-s + 0.361·55-s + ⋯

Functional equation

Λ(s)=(616s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(616s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 616616    =    23⋅7⋅112^{3} \cdot 7 \cdot 11
Sign: 11
Analytic conductor: 4.918784.91878
Root analytic conductor: 2.217832.21783
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 616, ( :1/2), 1)(2,\ 616,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.8413565212.841356521
L(12)L(\frac12) ≈\approx 2.8413565212.841356521
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
7 1−T 1 - T
11 1−T 1 - T
good3 1−2.68T+3T2 1 - 2.68T + 3T^{2}
5 1−2.68T+5T2 1 - 2.68T + 5T^{2}
13 1+2.50T+13T2 1 + 2.50T + 13T^{2}
17 1+6.37T+17T2 1 + 6.37T + 17T^{2}
19 1+3.87T+19T2 1 + 3.87T + 19T^{2}
23 1+4.55T+23T2 1 + 4.55T + 23T^{2}
29 1−3.01T+29T2 1 - 3.01T + 29T^{2}
31 1−5.18T+31T2 1 - 5.18T + 31T^{2}
37 1+6.55T+37T2 1 + 6.55T + 37T^{2}
41 1+4.34T+41T2 1 + 4.34T + 41T^{2}
43 1+1.01T+43T2 1 + 1.01T + 43T^{2}
47 1−0.637T+47T2 1 - 0.637T + 47T^{2}
53 1+3.01T+53T2 1 + 3.01T + 53T^{2}
59 1−12.0T+59T2 1 - 12.0T + 59T^{2}
61 1−15.6T+61T2 1 - 15.6T + 61T^{2}
67 1−5.56T+67T2 1 - 5.56T + 67T^{2}
71 1−11.5T+71T2 1 - 11.5T + 71T^{2}
73 1+13.3T+73T2 1 + 13.3T + 73T^{2}
79 1−6.37T+79T2 1 - 6.37T + 79T^{2}
83 1−2.50T+83T2 1 - 2.50T + 83T^{2}
89 1+12.9T+89T2 1 + 12.9T + 89T^{2}
97 1+10.8T+97T2 1 + 10.8T + 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

−10.17200699564353002570535707285, −9.767410128491079068465780218256, −8.698592453568943557245085899711, −8.404078271664583362594909700066, −7.11304303166610787336314853109, −6.28821758971901757156237277195, −4.92633633218914971899591074776, −3.88883278231138769830239128868, −2.43611974768842675042505582867, −1.96478652796702649110347377046, 1.96478652796702649110347377046, 2.43611974768842675042505582867, 3.88883278231138769830239128868, 4.92633633218914971899591074776, 6.28821758971901757156237277195, 7.11304303166610787336314853109, 8.404078271664583362594909700066, 8.698592453568943557245085899711, 9.767410128491079068465780218256, 10.17200699564353002570535707285

Graph of the ZZ-function along the critical line