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Properties

Label 4-525e2-1.1-c1e2-0-15
Degree 44
Conductor 275625275625
Sign 11
Analytic cond. 17.574017.5740
Root an. cond. 2.047472.04747
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  + 2·3-s + 4-s − 2·7-s + 3·9-s + 4·11-s + 2·12-s − 3·16-s + 4·17-s + 4·19-s − 4·21-s − 8·23-s + 4·27-s − 2·28-s − 4·29-s + 12·31-s + 8·33-s + 3·36-s − 4·37-s − 4·41-s + 4·44-s − 8·47-s − 6·48-s + 3·49-s + 8·51-s + 16·53-s + 8·57-s − 4·61-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s − 0.755·7-s + 9-s + 1.20·11-s + 0.577·12-s − 3/4·16-s + 0.970·17-s + 0.917·19-s − 0.872·21-s − 1.66·23-s + 0.769·27-s − 0.377·28-s − 0.742·29-s + 2.15·31-s + 1.39·33-s + 1/2·36-s − 0.657·37-s − 0.624·41-s + 0.603·44-s − 1.16·47-s − 0.866·48-s + 3/7·49-s + 1.12·51-s + 2.19·53-s + 1.05·57-s − 0.512·61-s + ⋯

Functional equation

Λ(s)=(275625s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(275625s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 275625275625    =    32⋅54⋅723^{2} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 17.574017.5740
Root analytic conductor: 2.047472.04747
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 275625, ( :1/2,1/2), 1)(4,\ 275625,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.1414919243.141491924
L(12)L(\frac12) ≈\approx 3.1414919243.141491924
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)
bad3C1C_1 (1−T)2 ( 1 - T )^{2}
5 1 1
7C1C_1 (1+T)2 ( 1 + T )^{2}
good2C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
11C4C_4 1−4T+6T2−4pT3+p2T4 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4}
13C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19D4D_{4} 1−4T+22T2−4pT3+p2T4 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4}
23C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
29C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
31D4D_{4} 1−12T+78T2−12pT3+p2T4 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+4T−2T2+4pT3+p2T4 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4}
41C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
43C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
47D4D_{4} 1+8T+30T2+8pT3+p2T4 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−16T+150T2−16pT3+p2T4 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
61C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
67C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
71D4D_{4} 1−20T+222T2−20pT3+p2T4 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4}
73D4D_{4} 1−16T+190T2−16pT3+p2T4 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−8T+94T2−8pT3+p2T4 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−16T+150T2−16pT3+p2T4 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4}
89C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
97D4D_{4} 1+8T+190T2+8pT3+p2T4 1 + 8 T + 190 T^{2} + 8 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

−11.28299398728800897049558528258, −10.39010313504787535454496522042, −9.950960302441261767004383990112, −9.868084733126321368304859486514, −9.269254249531983603936081662591, −9.022742741780680793134051674963, −8.302662917073549728740059517430, −7.995355329640417161264884598639, −7.63336443622346506727556439854, −6.78153876579360587710267337064, −6.71373771480295834420418654362, −6.28769649091711699777333965213, −5.48696862629181682043759596209, −4.97165091885064165462761536384, −4.05644795091330065926893331078, −3.75937823312199115821787225357, −3.29003001980827580184157311209, −2.52400202601718389199352352664, −2.00819675316936200586395952175, −1.05542103816238995373673142966, 1.05542103816238995373673142966, 2.00819675316936200586395952175, 2.52400202601718389199352352664, 3.29003001980827580184157311209, 3.75937823312199115821787225357, 4.05644795091330065926893331078, 4.97165091885064165462761536384, 5.48696862629181682043759596209, 6.28769649091711699777333965213, 6.71373771480295834420418654362, 6.78153876579360587710267337064, 7.63336443622346506727556439854, 7.995355329640417161264884598639, 8.302662917073549728740059517430, 9.022742741780680793134051674963, 9.269254249531983603936081662591, 9.868084733126321368304859486514, 9.950960302441261767004383990112, 10.39010313504787535454496522042, 11.28299398728800897049558528258

Graph of the ZZ-function along the critical line