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Properties

Label 4-1095200-1.1-c1e2-0-10
Degree 44
Conductor 10952001095200
Sign −1-1
Analytic cond. 69.830969.8309
Root an. cond. 2.890752.89075
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 8-s + 4·9-s − 10-s − 3·13-s + 16-s − 4·18-s + 20-s − 4·25-s + 3·26-s − 32-s + 4·36-s − 8·37-s − 40-s − 9·41-s + 4·45-s + 5·49-s + 4·50-s − 3·52-s − 7·53-s + 5·61-s + 64-s − 3·65-s − 4·72-s + 8·74-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s + 4/3·9-s − 0.316·10-s − 0.832·13-s + 1/4·16-s − 0.942·18-s + 0.223·20-s − 4/5·25-s + 0.588·26-s − 0.176·32-s + 2/3·36-s − 1.31·37-s − 0.158·40-s − 1.40·41-s + 0.596·45-s + 5/7·49-s + 0.565·50-s − 0.416·52-s − 0.961·53-s + 0.640·61-s + 1/8·64-s − 0.372·65-s − 0.471·72-s + 0.929·74-s + ⋯

Functional equation

Λ(s)=(1095200s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1095200s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 10952001095200    =    25⋅52⋅3722^{5} \cdot 5^{2} \cdot 37^{2}
Sign: −1-1
Analytic conductor: 69.830969.8309
Root analytic conductor: 2.890752.89075
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 1095200, ( :1/2,1/2), −1)(4,\ 1095200,\ (\ :1/2, 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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C1C_1 1+T 1 + T
5C2C_2 1−T+pT2 1 - T + p T^{2}
37C2C_2 1+8T+pT2 1 + 8 T + p T^{2}
good3C22C_2^2 1−4T2+p2T4 1 - 4 T^{2} + p^{2} T^{4}
7C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
11C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
13C2C_2 (1−2T+pT2)(1+5T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} )
17C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
19C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
23C22C_2^2 1−19T2+p2T4 1 - 19 T^{2} + p^{2} T^{4}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C22C_2^2 1+3T2+p2T4 1 + 3 T^{2} + p^{2} T^{4}
41C2C_2×\timesC2C_2 (1+2T+pT2)(1+7T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} )
43C22C_2^2 1+40T2+p2T4 1 + 40 T^{2} + p^{2} T^{4}
47C22C_2^2 1−23T2+p2T4 1 - 23 T^{2} + p^{2} T^{4}
53C2C_2×\timesC2C_2 (1−4T+pT2)(1+11T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} )
59C22C_2^2 1−104T2+p2T4 1 - 104 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1−13T+pT2)(1+8T+pT2) ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} )
67C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
71C22C_2^2 1+15T2+p2T4 1 + 15 T^{2} + p^{2} T^{4}
73C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
79C22C_2^2 1−17T2+p2T4 1 - 17 T^{2} + p^{2} T^{4}
83C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1+pT2)(1+6T+pT2) ( 1 + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
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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

−7.72046659580019593313543126522, −7.44233739171997135907955976169, −7.10477651159503319675021511502, −6.59418048139677526840874061479, −6.32835214985814926083945823243, −5.58984240437669804435224876506, −5.22759714613272812709239763260, −4.73085134156516446637164026689, −4.15671404059170282010540179471, −3.63536360159906566313247530381, −3.03424934942362814097272611877, −2.24854289456122829340209902841, −1.82870692585144507262393023918, −1.20507569496527294566002758393, 0, 1.20507569496527294566002758393, 1.82870692585144507262393023918, 2.24854289456122829340209902841, 3.03424934942362814097272611877, 3.63536360159906566313247530381, 4.15671404059170282010540179471, 4.73085134156516446637164026689, 5.22759714613272812709239763260, 5.58984240437669804435224876506, 6.32835214985814926083945823243, 6.59418048139677526840874061479, 7.10477651159503319675021511502, 7.44233739171997135907955976169, 7.72046659580019593313543126522

Graph of the ZZ-function along the critical line