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Properties

Label 4-525e2-1.1-c1e2-0-27
Degree 44
Conductor 275625275625
Sign −1-1
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 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·7-s − 2·9-s + 2·13-s − 4·16-s − 2·21-s − 5·27-s + 4·31-s + 4·37-s + 2·39-s − 8·43-s − 4·48-s + 3·49-s − 16·61-s + 4·63-s + 4·67-s + 12·73-s − 10·79-s + 81-s − 4·91-s + 4·93-s + 14·97-s − 38·103-s + 10·109-s + 4·111-s + 8·112-s − 4·117-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s − 2/3·9-s + 0.554·13-s − 16-s − 0.436·21-s − 0.962·27-s + 0.718·31-s + 0.657·37-s + 0.320·39-s − 1.21·43-s − 0.577·48-s + 3/7·49-s − 2.04·61-s + 0.503·63-s + 0.488·67-s + 1.40·73-s − 1.12·79-s + 1/9·81-s − 0.419·91-s + 0.414·93-s + 1.42·97-s − 3.74·103-s + 0.957·109-s + 0.379·111-s + 0.755·112-s − 0.369·117-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: −1-1
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: 11
Selberg data: (4, 275625, ( :1/2,1/2), −1)(4,\ 275625,\ (\ :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)
bad3C2C_2 1−T+pT2 1 - T + p T^{2}
5 1 1
7C1C_1 (1+T)2 ( 1 + T )^{2}
good2C2C_2 (1−pT+pT2)(1+pT+pT2) ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} )
11C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
13C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
17C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
31C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
37C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
41C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
61C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
67C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
71C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
73C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
79C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
83C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{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

−8.569343753087953468815992872802, −8.428704975511724884581109770971, −7.67053504463822546742181996665, −7.38451703018051703490900915946, −6.59661422823789620490429923129, −6.33345798756843332553352393681, −5.93898740434763322465740513982, −5.14936161735373762545181523253, −4.74217082052326318941224012916, −3.90430059874868075785507099311, −3.57752582126466520607951640819, −2.76472422088648867371864611007, −2.48112200668525509199201292270, −1.41698761693361156521107028690, 0, 1.41698761693361156521107028690, 2.48112200668525509199201292270, 2.76472422088648867371864611007, 3.57752582126466520607951640819, 3.90430059874868075785507099311, 4.74217082052326318941224012916, 5.14936161735373762545181523253, 5.93898740434763322465740513982, 6.33345798756843332553352393681, 6.59661422823789620490429923129, 7.38451703018051703490900915946, 7.67053504463822546742181996665, 8.428704975511724884581109770971, 8.569343753087953468815992872802

Graph of the ZZ-function along the critical line