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Properties

Label 2-460e2-1.1-c1-0-39
Degree 22
Conductor 211600211600
Sign −1-1
Analytic cond. 1689.631689.63
Root an. cond. 41.105141.1051
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  − 7-s − 3·9-s − 11-s − 13-s − 5·19-s − 5·29-s + 2·31-s − 4·37-s − 5·41-s + 9·43-s − 6·47-s − 6·49-s + 2·53-s − 8·59-s + 8·61-s + 3·63-s − 8·67-s + 10·71-s + 3·73-s + 77-s − 3·79-s + 9·81-s − 3·83-s − 10·89-s + 91-s − 2·97-s + 3·99-s + ⋯
L(s)  = 1  − 0.377·7-s − 9-s − 0.301·11-s − 0.277·13-s − 1.14·19-s − 0.928·29-s + 0.359·31-s − 0.657·37-s − 0.780·41-s + 1.37·43-s − 0.875·47-s − 6/7·49-s + 0.274·53-s − 1.04·59-s + 1.02·61-s + 0.377·63-s − 0.977·67-s + 1.18·71-s + 0.351·73-s + 0.113·77-s − 0.337·79-s + 81-s − 0.329·83-s − 1.05·89-s + 0.104·91-s − 0.203·97-s + 0.301·99-s + ⋯

Functional equation

Λ(s)=(211600s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(211600s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 211600211600    =    24⋅52⋅2322^{4} \cdot 5^{2} \cdot 23^{2}
Sign: −1-1
Analytic conductor: 1689.631689.63
Root analytic conductor: 41.105141.1051
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 211600, ( :1/2), −1)(2,\ 211600,\ (\ :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)
bad2 1 1
5 1 1
23 1 1
good3 1+pT2 1 + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+5T+pT2 1 + 5 T + p T^{2}
29 1+5T+pT2 1 + 5 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1+5T+pT2 1 + 5 T + p T^{2}
43 1−9T+pT2 1 - 9 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1−10T+pT2 1 - 10 T + p T^{2}
73 1−3T+pT2 1 - 3 T + p T^{2}
79 1+3T+pT2 1 + 3 T + p T^{2}
83 1+3T+pT2 1 + 3 T + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1+2T+pT2 1 + 2 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

−13.24213886110817, −12.72506834467972, −12.39674368693017, −11.83780307055722, −11.30686290064377, −10.97903800565240, −10.46141701980604, −9.984636602165827, −9.452660338151083, −8.999265062556307, −8.478723930796937, −8.145805353982929, −7.540183220183239, −7.037695333060061, −6.370945648504177, −6.151634162401228, −5.431848780609729, −5.101323888356897, −4.411305136047033, −3.864158118323200, −3.239594729593786, −2.789540649366389, −2.157953154324139, −1.636340283496817, −0.5765472876696678, 0, 0.5765472876696678, 1.636340283496817, 2.157953154324139, 2.789540649366389, 3.239594729593786, 3.864158118323200, 4.411305136047033, 5.101323888356897, 5.431848780609729, 6.151634162401228, 6.370945648504177, 7.037695333060061, 7.540183220183239, 8.145805353982929, 8.478723930796937, 8.999265062556307, 9.452660338151083, 9.984636602165827, 10.46141701980604, 10.97903800565240, 11.30686290064377, 11.83780307055722, 12.39674368693017, 12.72506834467972, 13.24213886110817

Graph of the ZZ-function along the critical line