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Properties

Label 2-460e2-1.1-c1-0-102
Degree 22
Conductor 211600211600
Sign 11
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 22

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·7-s − 2·9-s − 3·11-s − 4·13-s + 3·17-s + 5·19-s − 2·21-s + 5·27-s − 2·31-s + 3·33-s − 2·37-s + 4·39-s − 3·41-s − 4·43-s − 12·47-s − 3·49-s − 3·51-s − 6·53-s − 5·57-s − 2·61-s − 4·63-s − 13·67-s − 12·71-s + 11·73-s − 6·77-s − 10·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s − 2/3·9-s − 0.904·11-s − 1.10·13-s + 0.727·17-s + 1.14·19-s − 0.436·21-s + 0.962·27-s − 0.359·31-s + 0.522·33-s − 0.328·37-s + 0.640·39-s − 0.468·41-s − 0.609·43-s − 1.75·47-s − 3/7·49-s − 0.420·51-s − 0.824·53-s − 0.662·57-s − 0.256·61-s − 0.503·63-s − 1.58·67-s − 1.42·71-s + 1.28·73-s − 0.683·77-s − 1.12·79-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: 11
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: 22
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+T+pT2 1 + T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1−5T+pT2 1 - 5 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+3T+pT2 1 + 3 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+13T+pT2 1 + 13 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1−11T+pT2 1 - 11 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1+9T+pT2 1 + 9 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

−13.52942700993099, −12.87527705562080, −12.49658666138221, −11.89694791432739, −11.71386200410520, −11.16784420815525, −10.80194751428602, −10.15892727859441, −9.823728547444456, −9.406817243610037, −8.539387956362455, −8.331192549649400, −7.681753343161279, −7.407584311457936, −6.832752080773301, −6.144951571152375, −5.630706619020301, −5.109750871138102, −5.000560086348682, −4.387930737898722, −3.448920776196527, −2.983701574882919, −2.574271985728701, −1.643246633313582, −1.255262075652764, 0, 0, 1.255262075652764, 1.643246633313582, 2.574271985728701, 2.983701574882919, 3.448920776196527, 4.387930737898722, 5.000560086348682, 5.109750871138102, 5.630706619020301, 6.144951571152375, 6.832752080773301, 7.407584311457936, 7.681753343161279, 8.331192549649400, 8.539387956362455, 9.406817243610037, 9.823728547444456, 10.15892727859441, 10.80194751428602, 11.16784420815525, 11.71386200410520, 11.89694791432739, 12.49658666138221, 12.87527705562080, 13.52942700993099

Graph of the ZZ-function along the critical line