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Properties

Label 4-855e2-1.1-c1e2-0-16
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  − 2·2-s + 4-s + 2·5-s − 4·10-s − 4·11-s − 4·13-s + 16-s − 8·17-s − 2·19-s + 2·20-s + 8·22-s − 4·23-s + 3·25-s + 8·26-s − 12·31-s + 2·32-s + 16·34-s − 4·37-s + 4·38-s − 8·41-s + 16·43-s − 4·44-s + 8·46-s + 4·47-s − 12·49-s − 6·50-s − 4·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 0.894·5-s − 1.26·10-s − 1.20·11-s − 1.10·13-s + 1/4·16-s − 1.94·17-s − 0.458·19-s + 0.447·20-s + 1.70·22-s − 0.834·23-s + 3/5·25-s + 1.56·26-s − 2.15·31-s + 0.353·32-s + 2.74·34-s − 0.657·37-s + 0.648·38-s − 1.24·41-s + 2.43·43-s − 0.603·44-s + 1.17·46-s + 0.583·47-s − 1.71·49-s − 0.848·50-s − 0.554·52-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}
good2D4D_{4} 1+pT+3T2+p2T3+p2T4 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4}
7C22C_2^2 1+12T2+p2T4 1 + 12 T^{2} + p^{2} T^{4}
11C22C_2^2 1+4T+8T2+4pT3+p2T4 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}
13D4D_{4} 1+4T+28T2+4pT3+p2T4 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4}
17D4D_{4} 1+8T+42T2+8pT3+p2T4 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+4T+18T2+4pT3+p2T4 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+56T2+p2T4 1 + 56 T^{2} + p^{2} T^{4}
31D4D_{4} 1+12T+90T2+12pT3+p2T4 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+4T+76T2+4pT3+p2T4 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+8T+80T2+8pT3+p2T4 1 + 8 T + 80 T^{2} + 8 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−16T+132T2−16pT3+p2T4 1 - 16 T + 132 T^{2} - 16 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−4T+66T2−4pT3+p2T4 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4}
53C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
59D4D_{4} 1+8T+62T2+8pT3+p2T4 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+8T+10T2+8pT3+p2T4 1 + 8 T + 10 T^{2} + 8 p T^{3} + p^{2} T^{4}
67D4D_{4} 1+8T+118T2+8pT3+p2T4 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−16T+198T2−16pT3+p2T4 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+4T+118T2+4pT3+p2T4 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83D4D_{4} 1+20T+258T2+20pT3+p2T4 1 + 20 T + 258 T^{2} + 20 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−8T+96T2−8pT3+p2T4 1 - 8 T + 96 T^{2} - 8 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+28T+372T2+28pT3+p2T4 1 + 28 T + 372 T^{2} + 28 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

−9.722745648379049918646211507621, −9.514531458906954383141396676411, −9.227281111221382192476111583475, −8.866871035962801030941946926719, −8.214413845173848281060607772219, −8.187536557251497965678579957603, −7.37133136499828317476175292112, −7.25258869154653640498590323148, −6.59797370219109302594742518380, −6.04254267696844185646324438513, −5.76376858388568670659294503497, −5.03561476211193453184203592506, −4.69800476778969242459409458966, −4.19892764820537397559298863593, −3.24134624149838719801586204400, −2.66266741777911759740479304679, −2.06707568249052481512345127651, −1.65084733236590813945970926393, 0, 0, 1.65084733236590813945970926393, 2.06707568249052481512345127651, 2.66266741777911759740479304679, 3.24134624149838719801586204400, 4.19892764820537397559298863593, 4.69800476778969242459409458966, 5.03561476211193453184203592506, 5.76376858388568670659294503497, 6.04254267696844185646324438513, 6.59797370219109302594742518380, 7.25258869154653640498590323148, 7.37133136499828317476175292112, 8.187536557251497965678579957603, 8.214413845173848281060607772219, 8.866871035962801030941946926719, 9.227281111221382192476111583475, 9.514531458906954383141396676411, 9.722745648379049918646211507621

Graph of the ZZ-function along the critical line