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Properties

Label 2-39600-1.1-c1-0-8
Degree 22
Conductor 3960039600
Sign 11
Analytic cond. 316.207316.207
Root an. cond. 17.782217.7822
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  + 4·7-s − 11-s − 6·13-s − 6·17-s − 4·19-s − 4·23-s + 2·29-s − 8·31-s + 10·37-s − 10·41-s − 4·47-s + 9·49-s − 10·53-s − 4·59-s − 2·61-s − 8·67-s + 14·73-s − 4·77-s + 16·79-s + 8·83-s + 6·89-s − 24·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 1.51·7-s − 0.301·11-s − 1.66·13-s − 1.45·17-s − 0.917·19-s − 0.834·23-s + 0.371·29-s − 1.43·31-s + 1.64·37-s − 1.56·41-s − 0.583·47-s + 9/7·49-s − 1.37·53-s − 0.520·59-s − 0.256·61-s − 0.977·67-s + 1.63·73-s − 0.455·77-s + 1.80·79-s + 0.878·83-s + 0.635·89-s − 2.51·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

Λ(s)=(39600s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(39600s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3960039600    =    24⋅32⋅52⋅112^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11
Sign: 11
Analytic conductor: 316.207316.207
Root analytic conductor: 17.782217.7822
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 39600, ( :1/2), 1)(2,\ 39600,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1544966001.154496600
L(12)L(\frac12) ≈\approx 1.1544966001.154496600
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 1
11 1+T 1 + T
good7 1−4T+pT2 1 - 4 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+8T+pT2 1 + 8 T + 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+pT2 1 + p T^{2}
47 1+4T+pT2 1 + 4 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+8T+pT2 1 + 8 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 1−8T+pT2 1 - 8 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

−14.93719500517563, −14.29795791633874, −13.92887039558798, −13.16042398025185, −12.76158068889969, −12.12019357967023, −11.64699512281116, −11.11436601366636, −10.69407827424018, −10.14586559947530, −9.418742314029574, −8.976176561292709, −8.276790932253892, −7.783348122340537, −7.490959488468550, −6.588041266368004, −6.218220542483476, −5.143659821510484, −4.910863805935629, −4.437072559458446, −3.715139352843885, −2.653556615434410, −2.080776909293105, −1.712876515326923, −0.3660211538825820, 0.3660211538825820, 1.712876515326923, 2.080776909293105, 2.653556615434410, 3.715139352843885, 4.437072559458446, 4.910863805935629, 5.143659821510484, 6.218220542483476, 6.588041266368004, 7.490959488468550, 7.783348122340537, 8.276790932253892, 8.976176561292709, 9.418742314029574, 10.14586559947530, 10.69407827424018, 11.11436601366636, 11.64699512281116, 12.12019357967023, 12.76158068889969, 13.16042398025185, 13.92887039558798, 14.29795791633874, 14.93719500517563

Graph of the ZZ-function along the critical line