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Properties

Label 4-525e2-1.1-c1e2-0-17
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  + 3·3-s − 2·4-s + 4·7-s + 6·9-s − 6·12-s + 6·19-s + 12·21-s + 9·27-s − 8·28-s − 15·31-s − 12·36-s + 11·37-s − 10·43-s + 9·49-s + 18·57-s + 27·61-s + 24·63-s + 8·64-s + 16·67-s − 3·73-s − 12·76-s − 17·79-s + 9·81-s − 24·84-s − 45·93-s + 27·103-s − 18·108-s + ⋯
L(s)  = 1  + 1.73·3-s − 4-s + 1.51·7-s + 2·9-s − 1.73·12-s + 1.37·19-s + 2.61·21-s + 1.73·27-s − 1.51·28-s − 2.69·31-s − 2·36-s + 1.80·37-s − 1.52·43-s + 9/7·49-s + 2.38·57-s + 3.45·61-s + 3.02·63-s + 64-s + 1.95·67-s − 0.351·73-s − 1.37·76-s − 1.91·79-s + 81-s − 2.61·84-s − 4.66·93-s + 2.66·103-s − 1.73·108-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 3.2875341413.287534141
L(12)L(\frac12) ≈\approx 3.2875341413.287534141
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)
bad3C2C_2 1−pT+pT2 1 - p T + p T^{2}
5 1 1
7C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
good2C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
11C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
19C2C_2 (1−7T+pT2)(1+T+pT2) ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} )
23C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
29C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
31C2C_2 (1+4T+pT2)(1+11T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} )
37C2C_2 (1−10T+pT2)(1−T+pT2) ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} )
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
47C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
53C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
59C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
61C2C_2 (1−14T+pT2)(1−13T+pT2) ( 1 - 14 T + p T^{2} )( 1 - 13 T + p T^{2} )
67C2C_2 (1−11T+pT2)(1−5T+pT2) ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} )
71C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
73C2C_2 (1−7T+pT2)(1+10T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C2C_2 (1+4T+pT2)(1+13T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} )
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
97C2C_2 (1−19T+pT2)(1+19T+pT2) ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{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

−11.20898337722182738029236542287, −10.49831278620155977143497500687, −9.859324735411588998911630704885, −9.704529820792939241649061447679, −9.235219852338695996066870610755, −8.790112898646503023663470037703, −8.426284183633319152442091242206, −8.142644917649185827675823322083, −7.63114268332235906291337470677, −7.23859822324037744774027550113, −6.86686745065389677772433983252, −5.79601223912743460586308788474, −5.24684148744016270174242739152, −4.92459321086285908392015516086, −4.28750309905247473683710449697, −3.76773103021466577869378495194, −3.41195056851713465102309223673, −2.44495539396809546843751774069, −1.95392133223678978025031114901, −1.10319902927824971887898493440, 1.10319902927824971887898493440, 1.95392133223678978025031114901, 2.44495539396809546843751774069, 3.41195056851713465102309223673, 3.76773103021466577869378495194, 4.28750309905247473683710449697, 4.92459321086285908392015516086, 5.24684148744016270174242739152, 5.79601223912743460586308788474, 6.86686745065389677772433983252, 7.23859822324037744774027550113, 7.63114268332235906291337470677, 8.142644917649185827675823322083, 8.426284183633319152442091242206, 8.790112898646503023663470037703, 9.235219852338695996066870610755, 9.704529820792939241649061447679, 9.859324735411588998911630704885, 10.49831278620155977143497500687, 11.20898337722182738029236542287

Graph of the ZZ-function along the critical line