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Properties

Label 2-72e2-1.1-c1-0-11
Degree 22
Conductor 51845184
Sign 11
Analytic cond. 41.394441.3944
Root an. cond. 6.433856.43385
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  − 2·7-s − 3·11-s − 2·13-s − 3·17-s − 19-s + 6·23-s − 5·25-s − 6·29-s + 4·31-s + 4·37-s + 9·41-s − 43-s + 6·47-s − 3·49-s − 12·53-s + 3·59-s − 8·61-s + 5·67-s + 12·71-s + 11·73-s + 6·77-s + 4·79-s + 12·83-s + 6·89-s + 4·91-s + 5·97-s − 14·103-s + ⋯
L(s)  = 1  − 0.755·7-s − 0.904·11-s − 0.554·13-s − 0.727·17-s − 0.229·19-s + 1.25·23-s − 25-s − 1.11·29-s + 0.718·31-s + 0.657·37-s + 1.40·41-s − 0.152·43-s + 0.875·47-s − 3/7·49-s − 1.64·53-s + 0.390·59-s − 1.02·61-s + 0.610·67-s + 1.42·71-s + 1.28·73-s + 0.683·77-s + 0.450·79-s + 1.31·83-s + 0.635·89-s + 0.419·91-s + 0.507·97-s − 1.37·103-s + ⋯

Functional equation

Λ(s)=(5184s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5184s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 51845184    =    26⋅342^{6} \cdot 3^{4}
Sign: 11
Analytic conductor: 41.394441.3944
Root analytic conductor: 6.433856.43385
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5184, ( :1/2), 1)(2,\ 5184,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1433719001.143371900
L(12)L(\frac12) ≈\approx 1.1433719001.143371900
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
3 1 1
good5 1+pT2 1 + 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+2T+pT2 1 + 2 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1−9T+pT2 1 - 9 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1−3T+pT2 1 - 3 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1−5T+pT2 1 - 5 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 1−4T+pT2 1 - 4 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−5T+pT2 1 - 5 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

−8.004874510471126629875326215800, −7.61550392042635002094679885373, −6.71782060930200301497522083770, −6.13537847248252222252378323029, −5.27398052675085301748339586603, −4.58847837490118018713372112605, −3.66068742781442559176084506010, −2.78620746794391482576765657021, −2.08674986394077854980553765067, −0.55030123589812757559608034041, 0.55030123589812757559608034041, 2.08674986394077854980553765067, 2.78620746794391482576765657021, 3.66068742781442559176084506010, 4.58847837490118018713372112605, 5.27398052675085301748339586603, 6.13537847248252222252378323029, 6.71782060930200301497522083770, 7.61550392042635002094679885373, 8.004874510471126629875326215800

Graph of the ZZ-function along the critical line