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Properties

Label 2-4140-1.1-c1-0-10
Degree 22
Conductor 41404140
Sign 11
Analytic cond. 33.058033.0580
Root an. cond. 5.749615.74961
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  + 5-s − 4·7-s + 6·11-s − 13-s + 2·19-s − 23-s + 25-s − 9·29-s + 5·31-s − 4·35-s + 2·37-s + 9·41-s − 4·43-s + 3·47-s + 9·49-s + 6·53-s + 6·55-s + 2·61-s − 65-s − 10·67-s + 3·71-s − 7·73-s − 24·77-s − 10·79-s + 12·83-s + 4·91-s + 2·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s + 1.80·11-s − 0.277·13-s + 0.458·19-s − 0.208·23-s + 1/5·25-s − 1.67·29-s + 0.898·31-s − 0.676·35-s + 0.328·37-s + 1.40·41-s − 0.609·43-s + 0.437·47-s + 9/7·49-s + 0.824·53-s + 0.809·55-s + 0.256·61-s − 0.124·65-s − 1.22·67-s + 0.356·71-s − 0.819·73-s − 2.73·77-s − 1.12·79-s + 1.31·83-s + 0.419·91-s + 0.205·95-s + ⋯

Functional equation

Λ(s)=(4140s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4140s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 41404140    =    22⋅32⋅5⋅232^{2} \cdot 3^{2} \cdot 5 \cdot 23
Sign: 11
Analytic conductor: 33.058033.0580
Root analytic conductor: 5.749615.74961
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4140, ( :1/2), 1)(2,\ 4140,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8282411621.828241162
L(12)L(\frac12) ≈\approx 1.8282411621.828241162
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
5 1−T 1 - T
23 1+T 1 + T
good7 1+4T+pT2 1 + 4 T + p T^{2}
11 1−6T+pT2 1 - 6 T + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
29 1+9T+pT2 1 + 9 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−9T+pT2 1 - 9 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−3T+pT2 1 - 3 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+10T+pT2 1 + 10 T + p T^{2}
71 1−3T+pT2 1 - 3 T + p T^{2}
73 1+7T+pT2 1 + 7 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1−8T+pT2 1 - 8 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

−8.675195224509421244567010783131, −7.44458669533489548487414918137, −6.92397468338108808019310143609, −6.11387858034595661701643360874, −5.81664617436965714998858914525, −4.51281009230397575705652247627, −3.74499491504031266853477766636, −3.05859762092563524808570267988, −1.96419494679653881817999121224, −0.77418837928818037905531196406, 0.77418837928818037905531196406, 1.96419494679653881817999121224, 3.05859762092563524808570267988, 3.74499491504031266853477766636, 4.51281009230397575705652247627, 5.81664617436965714998858914525, 6.11387858034595661701643360874, 6.92397468338108808019310143609, 7.44458669533489548487414918137, 8.675195224509421244567010783131

Graph of the ZZ-function along the critical line