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Properties

Label 2-105e2-1.1-c1-0-33
Degree 22
Conductor 1102511025
Sign −1-1
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 11

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: −1-1
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: 11
Selberg data: (2, 11025, ( :1/2), −1)(2,\ 11025,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)Isogeny Class over Fp\mathbf{F}_p
bad3 1 1
5 1 1
7 1 1
good2 1+pT2 1 + p T^{2} 1.2.a
11 1+pT2 1 + p T^{2} 1.11.a
13 1−7T+pT2 1 - 7 T + p T^{2} 1.13.ah
17 1+pT2 1 + p T^{2} 1.17.a
19 1+7T+pT2 1 + 7 T + p T^{2} 1.19.h
23 1+pT2 1 + p T^{2} 1.23.a
29 1+pT2 1 + p T^{2} 1.29.a
31 1+7T+pT2 1 + 7 T + p T^{2} 1.31.h
37 1−T+pT2 1 - T + p T^{2} 1.37.ab
41 1+pT2 1 + p T^{2} 1.41.a
43 1+5T+pT2 1 + 5 T + p T^{2} 1.43.f
47 1+pT2 1 + p T^{2} 1.47.a
53 1+pT2 1 + p T^{2} 1.53.a
59 1+pT2 1 + p T^{2} 1.59.a
61 1−14T+pT2 1 - 14 T + p T^{2} 1.61.ao
67 1+11T+pT2 1 + 11 T + p T^{2} 1.67.l
71 1+pT2 1 + p T^{2} 1.71.a
73 1−7T+pT2 1 - 7 T + p T^{2} 1.73.ah
79 1+13T+pT2 1 + 13 T + p T^{2} 1.79.n
83 1+pT2 1 + p T^{2} 1.83.a
89 1+pT2 1 + p T^{2} 1.89.a
97 1+14T+pT2 1 + 14 T + p T^{2} 1.97.o
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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

−16.71375988647405, −16.35846470714573, −15.59845818185305, −15.00084486652746, −14.49174960868833, −13.86758950985686, −13.25204381064757, −12.97124369243016, −12.39577256882688, −11.45644965958982, −10.96763192907953, −10.39934631052245, −9.759612467709412, −8.937942024296082, −8.599436641612125, −8.178799140872350, −7.264553790946686, −6.415844632298848, −5.903067051363179, −5.227559122189026, −4.328793953204915, −3.872653074450142, −3.224339270854155, −1.993602805120383, −1.121051020092006, 0, 1.121051020092006, 1.993602805120383, 3.224339270854155, 3.872653074450142, 4.328793953204915, 5.227559122189026, 5.903067051363179, 6.415844632298848, 7.264553790946686, 8.178799140872350, 8.599436641612125, 8.937942024296082, 9.759612467709412, 10.39934631052245, 10.96763192907953, 11.45644965958982, 12.39577256882688, 12.97124369243016, 13.25204381064757, 13.86758950985686, 14.49174960868833, 15.00084486652746, 15.59845818185305, 16.35846470714573, 16.71375988647405

Graph of the ZZ-function along the critical line