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Properties

Label 2-460e2-1.1-c1-0-103
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  + 3·3-s + 4·7-s + 6·9-s − 3·11-s + 6·13-s − 2·17-s − 19-s + 12·21-s + 9·27-s − 2·29-s − 10·31-s − 9·33-s − 8·37-s + 18·39-s + 5·41-s − 11·43-s + 2·47-s + 9·49-s − 6·51-s + 12·53-s − 3·57-s − 13·59-s + 8·61-s + 24·63-s + 3·67-s − 10·71-s + 2·73-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.51·7-s + 2·9-s − 0.904·11-s + 1.66·13-s − 0.485·17-s − 0.229·19-s + 2.61·21-s + 1.73·27-s − 0.371·29-s − 1.79·31-s − 1.56·33-s − 1.31·37-s + 2.88·39-s + 0.780·41-s − 1.67·43-s + 0.291·47-s + 9/7·49-s − 0.840·51-s + 1.64·53-s − 0.397·57-s − 1.69·59-s + 1.02·61-s + 3.02·63-s + 0.366·67-s − 1.18·71-s + 0.234·73-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−pT+pT2 1 - p T + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+10T+pT2 1 + 10 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1−5T+pT2 1 - 5 T + p T^{2}
43 1+11T+pT2 1 + 11 T + p T^{2}
47 1−2T+pT2 1 - 2 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1+13T+pT2 1 + 13 T + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1−3T+pT2 1 - 3 T + p T^{2}
71 1+10T+pT2 1 + 10 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1−6T+pT2 1 - 6 T + p T^{2}
83 1+13T+pT2 1 + 13 T + p T^{2}
89 1+15T+pT2 1 + 15 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

−13.34164220084157, −12.99628000558333, −12.52671243009133, −11.78808795898790, −11.23157129778723, −10.86155720496877, −10.50635322381602, −9.930523565295381, −9.231429723610256, −8.816319328395302, −8.477965185859284, −8.232112779427486, −7.667761240401344, −7.252847179440312, −6.789076102980390, −5.931312148563202, −5.367958790485908, −4.957609869706802, −4.093475423348215, −3.955644730019425, −3.320406891149038, −2.694742233168878, −2.100248745628870, −1.623537035407646, −1.262120688458976, 0, 1.262120688458976, 1.623537035407646, 2.100248745628870, 2.694742233168878, 3.320406891149038, 3.955644730019425, 4.093475423348215, 4.957609869706802, 5.367958790485908, 5.931312148563202, 6.789076102980390, 7.252847179440312, 7.667761240401344, 8.232112779427486, 8.477965185859284, 8.816319328395302, 9.231429723610256, 9.930523565295381, 10.50635322381602, 10.86155720496877, 11.23157129778723, 11.78808795898790, 12.52671243009133, 12.99628000558333, 13.34164220084157

Graph of the ZZ-function along the critical line