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Properties

Label 2-4600-1.1-c1-0-11
Degree 22
Conductor 46004600
Sign 11
Analytic cond. 36.731136.7311
Root an. cond. 6.060626.06062
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  − 7-s − 3·9-s − 6·11-s + 2·13-s + 3·17-s − 6·19-s − 23-s + 3·29-s − 3·31-s − 37-s + 9·41-s + 8·43-s − 4·47-s − 6·49-s − 53-s + 59-s + 8·61-s + 3·63-s + 7·67-s − 5·71-s + 6·73-s + 6·77-s + 9·81-s + 11·83-s + 4·89-s − 2·91-s − 6·97-s + ⋯
L(s)  = 1  − 0.377·7-s − 9-s − 1.80·11-s + 0.554·13-s + 0.727·17-s − 1.37·19-s − 0.208·23-s + 0.557·29-s − 0.538·31-s − 0.164·37-s + 1.40·41-s + 1.21·43-s − 0.583·47-s − 6/7·49-s − 0.137·53-s + 0.130·59-s + 1.02·61-s + 0.377·63-s + 0.855·67-s − 0.593·71-s + 0.702·73-s + 0.683·77-s + 81-s + 1.20·83-s + 0.423·89-s − 0.209·91-s − 0.609·97-s + ⋯

Functional equation

Λ(s)=(4600s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4600s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46004600    =    23⋅52⋅232^{3} \cdot 5^{2} \cdot 23
Sign: 11
Analytic conductor: 36.731136.7311
Root analytic conductor: 6.060626.06062
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4600, ( :1/2), 1)(2,\ 4600,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0519140831.051914083
L(12)L(\frac12) ≈\approx 1.0519140831.051914083
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+T 1 + T
good3 1+pT2 1 + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
11 1+6T+pT2 1 + 6 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+6T+pT2 1 + 6 T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1+T+pT2 1 + T + p T^{2}
41 1−9T+pT2 1 - 9 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1+T+pT2 1 + T + p T^{2}
59 1−T+pT2 1 - T + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1−7T+pT2 1 - 7 T + p T^{2}
71 1+5T+pT2 1 + 5 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−11T+pT2 1 - 11 T + p T^{2}
89 1−4T+pT2 1 - 4 T + p T^{2}
97 1+6T+pT2 1 + 6 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

−8.131821699878571714817457957038, −7.895907093455660908259346655755, −6.81651891786475834789657050361, −5.98644200654968908267147201428, −5.52851532679300588628541511212, −4.68254936968660354804320686566, −3.65382078628700195903836662624, −2.83193822481131617809599919331, −2.16496033805434674415326045680, −0.53727497379349778489510799132, 0.53727497379349778489510799132, 2.16496033805434674415326045680, 2.83193822481131617809599919331, 3.65382078628700195903836662624, 4.68254936968660354804320686566, 5.52851532679300588628541511212, 5.98644200654968908267147201428, 6.81651891786475834789657050361, 7.895907093455660908259346655755, 8.131821699878571714817457957038

Graph of the ZZ-function along the critical line