↑

Properties

Label 4-855e2-1.1-c1e2-0-17
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s − 5-s − 8·7-s − 6·11-s − 2·13-s + 6·17-s − 7·19-s − 2·20-s − 16·28-s − 3·29-s − 14·31-s + 8·35-s + 16·37-s − 6·41-s + 4·43-s − 12·44-s + 6·47-s + 34·49-s − 4·52-s − 6·53-s + 6·55-s − 15·59-s − 5·61-s − 8·64-s + 2·65-s − 2·67-s + 12·68-s + ⋯
L(s)  = 1  + 4-s − 0.447·5-s − 3.02·7-s − 1.80·11-s − 0.554·13-s + 1.45·17-s − 1.60·19-s − 0.447·20-s − 3.02·28-s − 0.557·29-s − 2.51·31-s + 1.35·35-s + 2.63·37-s − 0.937·41-s + 0.609·43-s − 1.80·44-s + 0.875·47-s + 34/7·49-s − 0.554·52-s − 0.824·53-s + 0.809·55-s − 1.95·59-s − 0.640·61-s − 64-s + 0.248·65-s − 0.244·67-s + 1.45·68-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
5C2C_2 1+T+T2 1 + T + T^{2}
19C2C_2 1+7T+pT2 1 + 7 T + p T^{2}
good2C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
7C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
11C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
13C2C_2 (1−5T+pT2)(1+7T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} )
17C22C_2^2 1−6T+19T2−6pT3+p2T4 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
29C22C_2^2 1+3T−20T2+3pT3+p2T4 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4}
31C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
37C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
41C22C_2^2 1+6T−5T2+6pT3+p2T4 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4}
43C22C_2^2 1−4T−27T2−4pT3+p2T4 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−6T−11T2−6pT3+p2T4 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4}
53C22C_2^2 1+6T−17T2+6pT3+p2T4 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+15T+166T2+15pT3+p2T4 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4}
61C22C_2^2 1+5T−36T2+5pT3+p2T4 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+2T−63T2+2pT3+p2T4 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4}
71C22C_2^2 1+3T−62T2+3pT3+p2T4 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+8T−9T2+8pT3+p2T4 1 + 8 T - 9 T^{2} + 8 p T^{3} + p^{2} T^{4}
79C22C_2^2 1+5T−54T2+5pT3+p2T4 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4}
83C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
89C22C_2^2 1+15T+136T2+15pT3+p2T4 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4}
97C22C_2^2 1+8T−33T2+8pT3+p2T4 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4}
show more
show less
   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

−9.850530987711527256674952408026, −9.785501073806385524056213412322, −9.072292990423787685858194721242, −9.037732842181625587261620891294, −7.899332607614189981339704991996, −7.88682278198065847964142894844, −7.32401323469590469361745876697, −6.97271323239509172320534047471, −6.55010406227323394050438693908, −6.07952880579286187774227883498, −5.58331007864135834733462007136, −5.51980529134237890805706407274, −4.18387938804896131922298628918, −4.17893331521981998392333728543, −3.05477246548069317452286738383, −3.01920831431247113420676509838, −2.66935415676807917451504337439, −1.77095464773124816393736803881, 0, 0, 1.77095464773124816393736803881, 2.66935415676807917451504337439, 3.01920831431247113420676509838, 3.05477246548069317452286738383, 4.17893331521981998392333728543, 4.18387938804896131922298628918, 5.51980529134237890805706407274, 5.58331007864135834733462007136, 6.07952880579286187774227883498, 6.55010406227323394050438693908, 6.97271323239509172320534047471, 7.32401323469590469361745876697, 7.88682278198065847964142894844, 7.899332607614189981339704991996, 9.037732842181625587261620891294, 9.072292990423787685858194721242, 9.785501073806385524056213412322, 9.850530987711527256674952408026

Graph of the ZZ-function along the critical line