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Properties

Label 4-132300-1.1-c1e2-0-6
Degree 44
Conductor 132300132300
Sign 11
Analytic cond. 8.435568.43556
Root an. cond. 1.704231.70423
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4-s − 2·7-s + 9-s + 12-s − 4·13-s + 16-s + 8·19-s − 2·21-s + 25-s + 27-s − 2·28-s + 36-s + 12·37-s − 4·39-s − 8·43-s + 48-s + 3·49-s − 4·52-s + 8·57-s + 28·61-s − 2·63-s + 64-s − 24·67-s + 20·73-s + 75-s + 8·76-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s − 0.755·7-s + 1/3·9-s + 0.288·12-s − 1.10·13-s + 1/4·16-s + 1.83·19-s − 0.436·21-s + 1/5·25-s + 0.192·27-s − 0.377·28-s + 1/6·36-s + 1.97·37-s − 0.640·39-s − 1.21·43-s + 0.144·48-s + 3/7·49-s − 0.554·52-s + 1.05·57-s + 3.58·61-s − 0.251·63-s + 1/8·64-s − 2.93·67-s + 2.34·73-s + 0.115·75-s + 0.917·76-s + ⋯

Functional equation

Λ(s)=(132300s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(132300s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 132300132300    =    22⋅33⋅52⋅722^{2} \cdot 3^{3} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 8.435568.43556
Root analytic conductor: 1.704231.70423
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 132300, ( :1/2,1/2), 1)(4,\ 132300,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0418684302.041868430
L(12)L(\frac12) ≈\approx 2.0418684302.041868430
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)
bad2C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
3C1C_1 1−T 1 - T
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
7C1C_1 (1+T)2 ( 1 + T )^{2}
good11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
17C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
29C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
59C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
67C2C_2 (1+12T+pT2)2 ( 1 + 12 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 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79C2C_2 (1−16T+pT2)2 ( 1 - 16 T + p T^{2} )^{2}
83C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
97C2C_2 (1−2T+pT2)2 ( 1 - 2 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

−9.437870818355534879168031071986, −9.118954839776025628038155273468, −8.185634457079877915713969006222, −7.983671143656830403181428322838, −7.31970008287706573052341981271, −7.07941601012087388140383379015, −6.44305888127139518988332786048, −5.97975998469911468351052676550, −5.09599254318882530046900067399, −4.95725936124254114017607791343, −3.85422726721423423764474467794, −3.44893990823261081236809071688, −2.69678884580615531725956839791, −2.27051973415844153304852765225, −0.996088059894723745963619169753, 0.996088059894723745963619169753, 2.27051973415844153304852765225, 2.69678884580615531725956839791, 3.44893990823261081236809071688, 3.85422726721423423764474467794, 4.95725936124254114017607791343, 5.09599254318882530046900067399, 5.97975998469911468351052676550, 6.44305888127139518988332786048, 7.07941601012087388140383379015, 7.31970008287706573052341981271, 7.983671143656830403181428322838, 8.185634457079877915713969006222, 9.118954839776025628038155273468, 9.437870818355534879168031071986

Graph of the ZZ-function along the critical line