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Properties

Label 2-4368-1.1-c1-0-18
Degree 22
Conductor 43684368
Sign 11
Analytic cond. 34.878634.8786
Root an. cond. 5.905815.90581
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  + 3-s − 1.56·5-s − 7-s + 9-s + 2·11-s − 13-s − 1.56·15-s + 5.12·17-s + 2.43·19-s − 21-s − 4.68·23-s − 2.56·25-s + 27-s − 3.56·29-s − 1.56·31-s + 2·33-s + 1.56·35-s + 1.12·37-s − 39-s + 7.12·41-s + 9.56·43-s − 1.56·45-s + 6.68·47-s + 49-s + 5.12·51-s + 0.438·53-s − 3.12·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.698·5-s − 0.377·7-s + 0.333·9-s + 0.603·11-s − 0.277·13-s − 0.403·15-s + 1.24·17-s + 0.559·19-s − 0.218·21-s − 0.976·23-s − 0.512·25-s + 0.192·27-s − 0.661·29-s − 0.280·31-s + 0.348·33-s + 0.263·35-s + 0.184·37-s − 0.160·39-s + 1.11·41-s + 1.45·43-s − 0.232·45-s + 0.975·47-s + 0.142·49-s + 0.717·51-s + 0.0602·53-s − 0.421·55-s + ⋯

Functional equation

Λ(s)=(4368s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4368s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 43684368    =    24⋅3⋅7⋅132^{4} \cdot 3 \cdot 7 \cdot 13
Sign: 11
Analytic conductor: 34.878634.8786
Root analytic conductor: 5.905815.90581
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4368, ( :1/2), 1)(2,\ 4368,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0080118012.008011801
L(12)L(\frac12) ≈\approx 2.0080118012.008011801
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
7 1+T 1 + T
13 1+T 1 + T
good5 1+1.56T+5T2 1 + 1.56T + 5T^{2}
11 1−2T+11T2 1 - 2T + 11T^{2}
17 1−5.12T+17T2 1 - 5.12T + 17T^{2}
19 1−2.43T+19T2 1 - 2.43T + 19T^{2}
23 1+4.68T+23T2 1 + 4.68T + 23T^{2}
29 1+3.56T+29T2 1 + 3.56T + 29T^{2}
31 1+1.56T+31T2 1 + 1.56T + 31T^{2}
37 1−1.12T+37T2 1 - 1.12T + 37T^{2}
41 1−7.12T+41T2 1 - 7.12T + 41T^{2}
43 1−9.56T+43T2 1 - 9.56T + 43T^{2}
47 1−6.68T+47T2 1 - 6.68T + 47T^{2}
53 1−0.438T+53T2 1 - 0.438T + 53T^{2}
59 1+5.12T+59T2 1 + 5.12T + 59T^{2}
61 1+6T+61T2 1 + 6T + 61T^{2}
67 1−13.3T+67T2 1 - 13.3T + 67T^{2}
71 1−2.87T+71T2 1 - 2.87T + 71T^{2}
73 1+5.80T+73T2 1 + 5.80T + 73T^{2}
79 1−11.8T+79T2 1 - 11.8T + 79T^{2}
83 1+9.80T+83T2 1 + 9.80T + 83T^{2}
89 1−5.56T+89T2 1 - 5.56T + 89T^{2}
97 1+7.56T+97T2 1 + 7.56T + 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

−8.209197905554803173198414239762, −7.63440507596170888808173101697, −7.22429220934273708928997619655, −6.12296435506030564500514203359, −5.52373118515310058780871164410, −4.32082479291631640997077825386, −3.79083645770729656009810710698, −3.06498976679320005147287416152, −2.01780281347619080587613979567, −0.77977158341817037127408942396, 0.77977158341817037127408942396, 2.01780281347619080587613979567, 3.06498976679320005147287416152, 3.79083645770729656009810710698, 4.32082479291631640997077825386, 5.52373118515310058780871164410, 6.12296435506030564500514203359, 7.22429220934273708928997619655, 7.63440507596170888808173101697, 8.209197905554803173198414239762

Graph of the ZZ-function along the critical line