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Properties

Label 4-1110e2-1.1-c1e2-0-14
Degree 44
Conductor 12321001232100
Sign 11
Analytic cond. 78.559778.5597
Root an. cond. 2.977142.97714
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·5-s − 9-s + 10·11-s + 16-s − 2·20-s − 25-s + 6·29-s + 2·31-s + 36-s − 2·41-s − 10·44-s − 2·45-s + 13·49-s + 20·55-s + 16·59-s + 10·61-s − 64-s − 12·71-s + 2·80-s + 81-s − 16·89-s − 10·99-s + 100-s + 20·101-s − 2·109-s − 6·116-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.894·5-s − 1/3·9-s + 3.01·11-s + 1/4·16-s − 0.447·20-s − 1/5·25-s + 1.11·29-s + 0.359·31-s + 1/6·36-s − 0.312·41-s − 1.50·44-s − 0.298·45-s + 13/7·49-s + 2.69·55-s + 2.08·59-s + 1.28·61-s − 1/8·64-s − 1.42·71-s + 0.223·80-s + 1/9·81-s − 1.69·89-s − 1.00·99-s + 1/10·100-s + 1.99·101-s − 0.191·109-s − 0.557·116-s + ⋯

Functional equation

Λ(s)=(1232100s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1232100s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 12321001232100    =    22⋅32⋅52⋅3722^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2}
Sign: 11
Analytic conductor: 78.559778.5597
Root analytic conductor: 2.977142.97714
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1232100, ( :1/2,1/2), 1)(4,\ 1232100,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.0085218963.008521896
L(12)L(\frac12) ≈\approx 3.0085218963.008521896
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C2C_2 1+T2 1 + T^{2}
3C2C_2 1+T2 1 + T^{2}
5C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
37C2C_2 1+T2 1 + T^{2}
good7C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
11C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
13C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
17C22C_2^2 1−33T2+p2T4 1 - 33 T^{2} + p^{2} T^{4}
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
29C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
31C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
41C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
43C22C_2^2 1−37T2+p2T4 1 - 37 T^{2} + p^{2} T^{4}
47C22C_2^2 1−78T2+p2T4 1 - 78 T^{2} + p^{2} T^{4}
53C22C_2^2 1−97T2+p2T4 1 - 97 T^{2} + p^{2} T^{4}
59C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
61C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
67C22C_2^2 1−118T2+p2T4 1 - 118 T^{2} + p^{2} T^{4}
71C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
73C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C22C_2^2 1−102T2+p2T4 1 - 102 T^{2} + p^{2} T^{4}
89C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
97C22C_2^2 1−193T2+p2T4 1 - 193 T^{2} + p^{2} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.898806548509306387902157028838, −9.461932428575362119272997614415, −9.432365744164382493535185519442, −8.742533599501486685955799282245, −8.554724625608482015802809910950, −8.346948234449979839643579756319, −7.34736670098169758450464632897, −7.13639854461668622546552540165, −6.50567474779619146881879836596, −6.40974912693474023525917083813, −5.72328564930003353726158881728, −5.60384256760759524094402033659, −4.75956057505067696990028294807, −4.38541092871708392793850025358, −3.78019112165209341356467054099, −3.63997426744686948500345781353, −2.71631083381843678212150511518, −2.12256468552869514027262606471, −1.35581878153588960545147587834, −0.895888343365722567060548398489, 0.895888343365722567060548398489, 1.35581878153588960545147587834, 2.12256468552869514027262606471, 2.71631083381843678212150511518, 3.63997426744686948500345781353, 3.78019112165209341356467054099, 4.38541092871708392793850025358, 4.75956057505067696990028294807, 5.60384256760759524094402033659, 5.72328564930003353726158881728, 6.40974912693474023525917083813, 6.50567474779619146881879836596, 7.13639854461668622546552540165, 7.34736670098169758450464632897, 8.346948234449979839643579756319, 8.554724625608482015802809910950, 8.742533599501486685955799282245, 9.432365744164382493535185519442, 9.461932428575362119272997614415, 9.898806548509306387902157028838

Graph of the ZZ-function along the critical line