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Properties

Label 4-224e2-1.1-c1e2-0-27
Degree 44
Conductor 5017650176
Sign −1-1
Analytic cond. 3.199263.19926
Root an. cond. 1.337401.33740
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 6·9-s − 12·17-s − 6·25-s − 16·31-s + 4·41-s + 16·47-s + 3·49-s − 12·63-s + 16·71-s + 20·73-s − 32·79-s + 27·81-s − 12·89-s − 12·97-s + 32·103-s + 4·113-s − 24·119-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 72·153-s + 157-s + ⋯
L(s)  = 1  + 0.755·7-s − 2·9-s − 2.91·17-s − 6/5·25-s − 2.87·31-s + 0.624·41-s + 2.33·47-s + 3/7·49-s − 1.51·63-s + 1.89·71-s + 2.34·73-s − 3.60·79-s + 3·81-s − 1.27·89-s − 1.21·97-s + 3.15·103-s + 0.376·113-s − 2.20·119-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 5.82·153-s + 0.0798·157-s + ⋯

Functional equation

Λ(s)=(50176s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(50176s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5017650176    =    210⋅722^{10} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 3.199263.19926
Root analytic conductor: 1.337401.33740
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 50176, ( :1/2,1/2), −1)(4,\ 50176,\ (\ :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)
bad2 1 1
7C1C_1 (1−T)2 ( 1 - T )^{2}
good3C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
5C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
19C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
37C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
67C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{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 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C2C_2 (1+6T+pT2)2 ( 1 + 6 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

−9.688287404638284990728895668207, −9.117894034638208548514808203742, −8.749304120934004278681759852831, −8.560607806232820418165457845354, −7.79292083571821938290976650763, −7.26382021666693896035142333427, −6.67949194679031572918581545842, −5.82613210142107824134972732669, −5.69394971603405508863657314834, −4.92225264716551837532790929786, −4.18465932715971087267449515138, −3.60173413237973468362473708205, −2.40145156924384381758797773878, −2.14556791802447766709825303506, 0, 2.14556791802447766709825303506, 2.40145156924384381758797773878, 3.60173413237973468362473708205, 4.18465932715971087267449515138, 4.92225264716551837532790929786, 5.69394971603405508863657314834, 5.82613210142107824134972732669, 6.67949194679031572918581545842, 7.26382021666693896035142333427, 7.79292083571821938290976650763, 8.560607806232820418165457845354, 8.749304120934004278681759852831, 9.117894034638208548514808203742, 9.688287404638284990728895668207

Graph of the ZZ-function along the critical line