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Properties

Label 2-105e2-1.1-c1-0-5
Degree 22
Conductor 1102511025
Sign 11
Analytic cond. 88.035088.0350
Root an. cond. 9.382709.38270
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  − 2·4-s + 2·13-s + 4·16-s − 8·19-s + 7·31-s − 11·37-s − 5·43-s − 4·52-s + 61-s − 8·64-s + 16·67-s + 17·73-s + 16·76-s + 17·79-s − 19·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 4-s + 0.554·13-s + 16-s − 1.83·19-s + 1.25·31-s − 1.80·37-s − 0.762·43-s − 0.554·52-s + 0.128·61-s − 64-s + 1.95·67-s + 1.98·73-s + 1.83·76-s + 1.91·79-s − 1.92·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

Λ(s)=(11025s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(11025s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1102511025    =    32⋅52⋅723^{2} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 88.035088.0350
Root analytic conductor: 9.382709.38270
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 11025, ( :1/2), 1)(2,\ 11025,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1572140511.157214051
L(12)L(\frac12) ≈\approx 1.1572140511.157214051
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)
bad3 1 1
5 1 1
7 1 1
good2 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−7T+pT2 1 - 7 T + p T^{2}
37 1+11T+pT2 1 + 11 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+5T+pT2 1 + 5 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+pT2 1 + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−T+pT2 1 - T + p T^{2}
67 1−16T+pT2 1 - 16 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−17T+pT2 1 - 17 T + p T^{2}
79 1−17T+pT2 1 - 17 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+19T+pT2 1 + 19 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

−16.71714877920568, −15.85461877252992, −15.24792721216111, −14.86654960411351, −13.99485931529907, −13.76210028317134, −13.11108108909342, −12.48348278681751, −12.13708425747135, −11.14333733138538, −10.66354867121160, −10.04109888947759, −9.473812186944616, −8.637836266285948, −8.429203698586159, −7.798286048753875, −6.640940709654943, −6.417828232224431, −5.346933873496737, −4.923559575528694, −4.017938602216696, −3.656459207176328, −2.568476489958289, −1.605716865626998, −0.5121595037667615, 0.5121595037667615, 1.605716865626998, 2.568476489958289, 3.656459207176328, 4.017938602216696, 4.923559575528694, 5.346933873496737, 6.417828232224431, 6.640940709654943, 7.798286048753875, 8.429203698586159, 8.637836266285948, 9.473812186944616, 10.04109888947759, 10.66354867121160, 11.14333733138538, 12.13708425747135, 12.48348278681751, 13.11108108909342, 13.76210028317134, 13.99485931529907, 14.86654960411351, 15.24792721216111, 15.85461877252992, 16.71714877920568

Graph of the ZZ-function along the critical line