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Properties

Label 4-3528e2-1.1-c1e2-0-6
Degree 44
Conductor 1244678412446784
Sign 11
Analytic cond. 793.617793.617
Root an. cond. 5.307655.30765
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 4·11-s − 4·13-s + 2·17-s + 4·19-s − 8·23-s + 5·25-s − 12·29-s − 8·31-s − 6·37-s + 12·41-s + 8·43-s − 2·53-s − 8·55-s + 4·59-s + 2·61-s + 8·65-s + 4·67-s − 16·71-s − 10·73-s + 8·79-s + 8·83-s − 4·85-s − 6·89-s − 8·95-s + 4·97-s − 18·101-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.20·11-s − 1.10·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 25-s − 2.22·29-s − 1.43·31-s − 0.986·37-s + 1.87·41-s + 1.21·43-s − 0.274·53-s − 1.07·55-s + 0.520·59-s + 0.256·61-s + 0.992·65-s + 0.488·67-s − 1.89·71-s − 1.17·73-s + 0.900·79-s + 0.878·83-s − 0.433·85-s − 0.635·89-s − 0.820·95-s + 0.406·97-s − 1.79·101-s + ⋯

Functional equation

Λ(s)=(12446784s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(12446784s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1244678412446784    =    26⋅34⋅742^{6} \cdot 3^{4} \cdot 7^{4}
Sign: 11
Analytic conductor: 793.617793.617
Root analytic conductor: 5.307655.30765
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 12446784, ( :1/2,1/2), 1)(4,\ 12446784,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.54128652540.5412865254
L(12)L(\frac12) ≈\approx 0.54128652540.5412865254
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)
bad2 1 1
3 1 1
7 1 1
good5C22C_2^2 1+2T−T2+2pT3+p2T4 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−4T+5T2−4pT3+p2T4 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4}
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
17C22C_2^2 1−2T−13T2−2pT3+p2T4 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4}
19C22C_2^2 1−4T−3T2−4pT3+p2T4 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+8T+41T2+8pT3+p2T4 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4}
29C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
31C22C_2^2 1+8T+33T2+8pT3+p2T4 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4}
37C22C_2^2 1+6T−T2+6pT3+p2T4 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4}
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
47C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
53C22C_2^2 1+2T−49T2+2pT3+p2T4 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−4T−43T2−4pT3+p2T4 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−2T−57T2−2pT3+p2T4 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−4T−51T2−4pT3+p2T4 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4}
71C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
73C2C_2 (1−7T+pT2)(1+17T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} )
79C22C_2^2 1−8T−15T2−8pT3+p2T4 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4}
83C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
89C22C_2^2 1+6T−53T2+6pT3+p2T4 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4}
97C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{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

−9.204046596487589180186904514203, −8.272553344426047313612471424621, −7.85512773338797021093032314339, −7.59683149763123796779310282940, −7.45974219323980043518733786024, −6.90630717232796421988368871476, −6.76247297475051832343364194177, −5.96531850397358392662028126581, −5.84162205342835299652013332632, −5.24370443692848804349558117651, −5.16727025592672306048199659099, −4.25537204051607515531005631063, −4.14836954244510441850702376959, −3.66051715115320171014303449147, −3.56105981485770482564089602739, −2.57539903730649162076855502659, −2.51177880333869842261581254860, −1.54596178224669240159256437825, −1.31174783387325350265933982568, −0.22571247862950536842039322377, 0.22571247862950536842039322377, 1.31174783387325350265933982568, 1.54596178224669240159256437825, 2.51177880333869842261581254860, 2.57539903730649162076855502659, 3.56105981485770482564089602739, 3.66051715115320171014303449147, 4.14836954244510441850702376959, 4.25537204051607515531005631063, 5.16727025592672306048199659099, 5.24370443692848804349558117651, 5.84162205342835299652013332632, 5.96531850397358392662028126581, 6.76247297475051832343364194177, 6.90630717232796421988368871476, 7.45974219323980043518733786024, 7.59683149763123796779310282940, 7.85512773338797021093032314339, 8.272553344426047313612471424621, 9.204046596487589180186904514203

Graph of the ZZ-function along the critical line