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Properties

Label 4-1095200-1.1-c1e2-0-14
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 + 8-s + 4·9-s + 2·13-s + 16-s − 15·17-s + 4·18-s − 5·25-s + 2·26-s + 32-s − 15·34-s + 4·36-s + 2·41-s − 5·49-s − 5·50-s + 2·52-s + 5·53-s − 12·61-s + 64-s − 15·68-s + 4·72-s − 6·73-s + 7·81-s + 2·82-s − 21·89-s + 12·97-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 4/3·9-s + 0.554·13-s + 1/4·16-s − 3.63·17-s + 0.942·18-s − 25-s + 0.392·26-s + 0.176·32-s − 2.57·34-s + 2/3·36-s + 0.312·41-s − 5/7·49-s − 0.707·50-s + 0.277·52-s + 0.686·53-s − 1.53·61-s + 1/8·64-s − 1.81·68-s + 0.471·72-s − 0.702·73-s + 7/9·81-s + 0.220·82-s − 2.22·89-s + 1.21·97-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+pT2 1 + p T^{2}
37C2C_2 1+pT2 1 + p T^{2}
good3C22C_2^2 1−4T2+p2T4 1 - 4 T^{2} + p^{2} T^{4}
7C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
11C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1−4T+pT2)(1+2T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2×\timesC2C_2 (1+7T+pT2)(1+8T+pT2) ( 1 + 7 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
23C22C_2^2 1+16T2+p2T4 1 + 16 T^{2} + p^{2} T^{4}
29C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
31C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C2C_2×\timesC2C_2 (1−6T+pT2)(1+4T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} )
43C22C_2^2 1−59T2+p2T4 1 - 59 T^{2} + p^{2} T^{4}
47C22C_2^2 1+71T2+p2T4 1 + 71 T^{2} + p^{2} T^{4}
53C2C_2×\timesC2C_2 (1−10T+pT2)(1+5T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} )
59C22C_2^2 1+45T2+p2T4 1 + 45 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1−3T+pT2)(1+15T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 15 T + p T^{2} )
67C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
71C22C_2^2 1+25T2+p2T4 1 + 25 T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1−T+pT2)(1+7T+pT2) ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} )
79C22C_2^2 1−96T2+p2T4 1 - 96 T^{2} + p^{2} T^{4}
83C22C_2^2 1+74T2+p2T4 1 + 74 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1+10T+pT2)(1+11T+pT2) ( 1 + 10 T + p T^{2} )( 1 + 11 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−10T+pT2)(1−2T+pT2) ( 1 - 10 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

−7.85352427209644252529875878279, −7.12326560527005019959983901029, −6.99608647630267093889034553384, −6.50482716048609744203410300214, −6.20509494638406742355830022524, −5.67833703367200168513113101759, −4.98540935785926339874018105138, −4.51185622749849948715272631835, −4.20485736101064834491331654886, −3.97713276546627912326365603184, −3.16050582969469397645827641236, −2.43068626843707114376811070245, −1.98823024842304050204700776123, −1.39584241770973838484214809799, 0, 1.39584241770973838484214809799, 1.98823024842304050204700776123, 2.43068626843707114376811070245, 3.16050582969469397645827641236, 3.97713276546627912326365603184, 4.20485736101064834491331654886, 4.51185622749849948715272631835, 4.98540935785926339874018105138, 5.67833703367200168513113101759, 6.20509494638406742355830022524, 6.50482716048609744203410300214, 6.99608647630267093889034553384, 7.12326560527005019959983901029, 7.85352427209644252529875878279

Graph of the ZZ-function along the critical line