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Properties

Label 4-9702e2-1.1-c1e2-0-25
Degree 44
Conductor 9412880494128804
Sign 11
Analytic cond. 6001.736001.73
Root an. cond. 8.801758.80175
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 2·11-s + 5·16-s − 4·22-s − 12·23-s − 10·25-s + 8·29-s − 6·32-s + 4·37-s + 20·43-s + 6·44-s + 24·46-s + 20·50-s − 4·53-s − 16·58-s + 7·64-s + 16·67-s − 32·71-s − 8·74-s − 16·79-s − 40·86-s − 8·88-s − 36·92-s − 30·100-s + 8·106-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s + 0.603·11-s + 5/4·16-s − 0.852·22-s − 2.50·23-s − 2·25-s + 1.48·29-s − 1.06·32-s + 0.657·37-s + 3.04·43-s + 0.904·44-s + 3.53·46-s + 2.82·50-s − 0.549·53-s − 2.10·58-s + 7/8·64-s + 1.95·67-s − 3.79·71-s − 0.929·74-s − 1.80·79-s − 4.31·86-s − 0.852·88-s − 3.75·92-s − 3·100-s + 0.777·106-s + ⋯

Functional equation

Λ(s)=(94128804s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(94128804s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9412880494128804    =    22⋅34⋅74⋅1122^{2} \cdot 3^{4} \cdot 7^{4} \cdot 11^{2}
Sign: 11
Analytic conductor: 6001.736001.73
Root analytic conductor: 8.801758.80175
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 94128804, ( :1/2,1/2), 1)(4,\ 94128804,\ (\ :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)2 ( 1 + T )^{2}
3 1 1
7 1 1
11C1C_1 (1−T)2 ( 1 - T )^{2}
good5C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
17C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
19C22C_2^2 1+20T2+p2T4 1 + 20 T^{2} + p^{2} T^{4}
23C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
29C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
31C22C_2^2 1+12T2+p2T4 1 + 12 T^{2} + p^{2} T^{4}
37C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
41C22C_2^2 1+74T2+p2T4 1 + 74 T^{2} + p^{2} T^{4}
43C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
47C22C_2^2 1−68T2+p2T4 1 - 68 T^{2} + p^{2} T^{4}
53C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
59C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
61C22C_2^2 1+24T2+p2T4 1 + 24 T^{2} + p^{2} T^{4}
67C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
71C2C_2 (1+16T+pT2)2 ( 1 + 16 T + p T^{2} )^{2}
73C22C_2^2 1+74T2+p2T4 1 + 74 T^{2} + p^{2} T^{4}
79C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
83C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
89C22C_2^2 1+128T2+p2T4 1 + 128 T^{2} + p^{2} T^{4}
97C22C_2^2 1+144T2+p2T4 1 + 144 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

−7.61306098166435554070562429733, −7.49151797748353724339772904966, −6.73434738171616199134482069220, −6.65384527303565678162733941289, −6.15284543364006056548718299211, −5.96843709024867243502708045318, −5.67611566136597490879348210320, −5.35891266185092378158829619906, −4.53764696532617191512962115334, −4.30847609020242236869298969714, −3.87591841174532161644364408016, −3.81771704539733165727045683571, −2.91948815150303705673915894821, −2.73373241935686791236895631579, −2.19371925871591313739375559171, −1.96006000917457453118379825711, −1.22292499567795485421307076375, −1.10508380038814680699142894786, 0, 0, 1.10508380038814680699142894786, 1.22292499567795485421307076375, 1.96006000917457453118379825711, 2.19371925871591313739375559171, 2.73373241935686791236895631579, 2.91948815150303705673915894821, 3.81771704539733165727045683571, 3.87591841174532161644364408016, 4.30847609020242236869298969714, 4.53764696532617191512962115334, 5.35891266185092378158829619906, 5.67611566136597490879348210320, 5.96843709024867243502708045318, 6.15284543364006056548718299211, 6.65384527303565678162733941289, 6.73434738171616199134482069220, 7.49151797748353724339772904966, 7.61306098166435554070562429733

Graph of the ZZ-function along the critical line