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Properties

Label 2-105e2-1.1-c1-0-4
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 − 7·13-s + 4·16-s + 7·19-s + 7·31-s + 37-s − 5·43-s + 14·52-s − 14·61-s − 8·64-s − 11·67-s − 7·73-s − 14·76-s − 13·79-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 4-s − 1.94·13-s + 16-s + 1.60·19-s + 1.25·31-s + 0.164·37-s − 0.762·43-s + 1.94·52-s − 1.79·61-s − 64-s − 1.34·67-s − 0.819·73-s − 1.60·76-s − 1.46·79-s + 1.42·97-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 1.0815211131.081521113
L(12)L(\frac12) ≈\approx 1.0815211131.081521113
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+7T+pT2 1 + 7 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−7T+pT2 1 - 7 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−T+pT2 1 - 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+14T+pT2 1 + 14 T + p T^{2}
67 1+11T+pT2 1 + 11 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+7T+pT2 1 + 7 T + p T^{2}
79 1+13T+pT2 1 + 13 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+pT2 1 + p T^{2}
97 1−14T+pT2 1 - 14 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

−16.61258139223701, −15.96957203593803, −15.18580870696054, −14.77938123448080, −14.11791316820387, −13.75354103870537, −13.12279074842885, −12.43814889379538, −11.95908325149761, −11.52468544963293, −10.35940226029709, −10.05066267034014, −9.476358465367494, −8.994350389100878, −8.165360561047153, −7.564214068400154, −7.137679194840246, −6.100722498393278, −5.402551153050883, −4.763923625670569, −4.402141274703869, −3.260457125717232, −2.776775483143016, −1.562264018334010, −0.4974465266779612, 0.4974465266779612, 1.562264018334010, 2.776775483143016, 3.260457125717232, 4.402141274703869, 4.763923625670569, 5.402551153050883, 6.100722498393278, 7.137679194840246, 7.564214068400154, 8.165360561047153, 8.994350389100878, 9.476358465367494, 10.05066267034014, 10.35940226029709, 11.52468544963293, 11.95908325149761, 12.43814889379538, 13.12279074842885, 13.75354103870537, 14.11791316820387, 14.77938123448080, 15.18580870696054, 15.96957203593803, 16.61258139223701

Graph of the ZZ-function along the critical line