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Properties

Label 4-855e2-1.1-c1e2-0-15
Degree 44
Conductor 731025731025
Sign 11
Analytic cond. 46.610746.6107
Root an. cond. 2.612892.61289
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 2·5-s − 2·7-s − 6·11-s − 2·13-s − 3·16-s + 2·19-s + 2·20-s + 3·25-s + 2·28-s − 6·29-s + 4·31-s + 4·35-s − 2·37-s − 6·41-s − 2·43-s + 6·44-s − 8·49-s + 2·52-s + 12·53-s + 12·55-s − 12·59-s − 20·61-s + 7·64-s + 4·65-s + 16·67-s − 12·71-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.894·5-s − 0.755·7-s − 1.80·11-s − 0.554·13-s − 3/4·16-s + 0.458·19-s + 0.447·20-s + 3/5·25-s + 0.377·28-s − 1.11·29-s + 0.718·31-s + 0.676·35-s − 0.328·37-s − 0.937·41-s − 0.304·43-s + 0.904·44-s − 8/7·49-s + 0.277·52-s + 1.64·53-s + 1.61·55-s − 1.56·59-s − 2.56·61-s + 7/8·64-s + 0.496·65-s + 1.95·67-s − 1.42·71-s + ⋯

Functional equation

Λ(s)=(731025s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(731025s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 731025731025    =    34⋅52⋅1923^{4} \cdot 5^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 46.610746.6107
Root analytic conductor: 2.612892.61289
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 731025, ( :1/2,1/2), 1)(4,\ 731025,\ (\ :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)
bad3 1 1
5C1C_1 (1+T)2 ( 1 + T )^{2}
19C1C_1 (1−T)2 ( 1 - T )^{2}
good2C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
7D4D_{4} 1+2T+12T2+2pT3+p2T4 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4}
11D4D_{4} 1+6T+28T2+6pT3+p2T4 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4}
13D4D_{4} 1+2T+24T2+2pT3+p2T4 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4}
17C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
29D4D_{4} 1+6T+40T2+6pT3+p2T4 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−4T+18T2−4pT3+p2T4 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+2T+48T2+2pT3+p2T4 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+6T+88T2+6pT3+p2T4 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+2T+60T2+2pT3+p2T4 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4}
47C22C_2^2 1+82T2+p2T4 1 + 82 T^{2} + p^{2} T^{4}
53D4D_{4} 1−12T+130T2−12pT3+p2T4 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+12T+142T2+12pT3+p2T4 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+20T+210T2+20pT3+p2T4 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4}
67C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
71D4D_{4} 1+12T+70T2+12pT3+p2T4 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+20T+198T2+20pT3+p2T4 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+8T+126T2+8pT3+p2T4 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−12T+154T2−12pT3+p2T4 1 - 12 T + 154 T^{2} - 12 p T^{3} + p^{2} T^{4}
89D4D_{4} 1+18T+256T2+18pT3+p2T4 1 + 18 T + 256 T^{2} + 18 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+2T+168T2+2pT3+p2T4 1 + 2 T + 168 T^{2} + 2 p T^{3} + 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

−9.883189261924100285367699278865, −9.662192409485576623441819539167, −8.940822958268431086479763387500, −8.918811508542577170579471553856, −8.081630085440141152762217752635, −7.925284616352481666971539413070, −7.51962887526028270450858107400, −6.96919026304098461403641387292, −6.68577435659982868662434597231, −5.98955303734139928413255293136, −5.31448895197748011351559390899, −5.22356174996129912622679127077, −4.42657179225334757009567689386, −4.26098337197706662680269410074, −3.38757928029076444734609280301, −2.97400430415512322974018382064, −2.55116332079524321306653284472, −1.55618517111585788553635139092, 0, 0, 1.55618517111585788553635139092, 2.55116332079524321306653284472, 2.97400430415512322974018382064, 3.38757928029076444734609280301, 4.26098337197706662680269410074, 4.42657179225334757009567689386, 5.22356174996129912622679127077, 5.31448895197748011351559390899, 5.98955303734139928413255293136, 6.68577435659982868662434597231, 6.96919026304098461403641387292, 7.51962887526028270450858107400, 7.925284616352481666971539413070, 8.081630085440141152762217752635, 8.918811508542577170579471553856, 8.940822958268431086479763387500, 9.662192409485576623441819539167, 9.883189261924100285367699278865

Graph of the ZZ-function along the critical line