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Properties

Label 2-5376-1.1-c1-0-20
Degree 22
Conductor 53765376
Sign 11
Analytic cond. 42.927542.9275
Root an. cond. 6.551916.55191
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 − 0.922·5-s + 7-s + 9-s + 0.382·11-s − 4.13·13-s − 0.922·15-s − 7.05·17-s − 1.30·19-s + 21-s − 2.44·23-s − 4.14·25-s + 27-s + 5.30·29-s + 10.9·31-s + 0.382·33-s − 0.922·35-s + 10.1·37-s − 4.13·39-s − 4.44·41-s − 0.983·43-s − 0.922·45-s + 7.50·47-s + 49-s − 7.05·51-s + 7.14·53-s − 0.352·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.412·5-s + 0.377·7-s + 0.333·9-s + 0.115·11-s − 1.14·13-s − 0.238·15-s − 1.71·17-s − 0.299·19-s + 0.218·21-s − 0.510·23-s − 0.829·25-s + 0.192·27-s + 0.985·29-s + 1.96·31-s + 0.0665·33-s − 0.155·35-s + 1.67·37-s − 0.661·39-s − 0.694·41-s − 0.150·43-s − 0.137·45-s + 1.09·47-s + 0.142·49-s − 0.987·51-s + 0.982·53-s − 0.0475·55-s + ⋯

Functional equation

Λ(s)=(5376s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5376s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 53765376    =    28⋅3⋅72^{8} \cdot 3 \cdot 7
Sign: 11
Analytic conductor: 42.927542.9275
Root analytic conductor: 6.551916.55191
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5376, ( :1/2), 1)(2,\ 5376,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9856704961.985670496
L(12)L(\frac12) ≈\approx 1.9856704961.985670496
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
good5 1+0.922T+5T2 1 + 0.922T + 5T^{2}
11 1−0.382T+11T2 1 - 0.382T + 11T^{2}
13 1+4.13T+13T2 1 + 4.13T + 13T^{2}
17 1+7.05T+17T2 1 + 7.05T + 17T^{2}
19 1+1.30T+19T2 1 + 1.30T + 19T^{2}
23 1+2.44T+23T2 1 + 2.44T + 23T^{2}
29 1−5.30T+29T2 1 - 5.30T + 29T^{2}
31 1−10.9T+31T2 1 - 10.9T + 31T^{2}
37 1−10.1T+37T2 1 - 10.1T + 37T^{2}
41 1+4.44T+41T2 1 + 4.44T + 41T^{2}
43 1+0.983T+43T2 1 + 0.983T + 43T^{2}
47 1−7.50T+47T2 1 - 7.50T + 47T^{2}
53 1−7.14T+53T2 1 - 7.14T + 53T^{2}
59 1−11.5T+59T2 1 - 11.5T + 59T^{2}
61 1−8.32T+61T2 1 - 8.32T + 61T^{2}
67 1−6.51T+67T2 1 - 6.51T + 67T^{2}
71 1+8.86T+71T2 1 + 8.86T + 71T^{2}
73 1−9.50T+73T2 1 - 9.50T + 73T^{2}
79 1−3.81T+79T2 1 - 3.81T + 79T^{2}
83 1−12.2T+83T2 1 - 12.2T + 83T^{2}
89 1−5.97T+89T2 1 - 5.97T + 89T^{2}
97 1−4.42T+97T2 1 - 4.42T + 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

−8.247005579688466412410870419581, −7.57441147352036106096565269702, −6.80805189428463985011461315822, −6.21711672567293166425383732736, −5.03132343017723624145589833957, −4.41991128955583620286566928089, −3.86093293540110797451626506464, −2.53199765661831315403738181698, −2.24644370940313937654102794808, −0.71635797505961202000738950155, 0.71635797505961202000738950155, 2.24644370940313937654102794808, 2.53199765661831315403738181698, 3.86093293540110797451626506464, 4.41991128955583620286566928089, 5.03132343017723624145589833957, 6.21711672567293166425383732736, 6.80805189428463985011461315822, 7.57441147352036106096565269702, 8.247005579688466412410870419581

Graph of the ZZ-function along the critical line