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Properties

Label 4-43904-1.1-c1e2-0-17
Degree 44
Conductor 4390443904
Sign −1-1
Analytic cond. 2.799352.79935
Root an. cond. 1.293491.29349
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  + 7-s − 6·9-s − 16·19-s − 6·25-s + 12·29-s − 16·31-s − 4·37-s + 16·47-s + 49-s + 12·53-s − 6·63-s + 27·81-s − 16·83-s + 32·103-s − 20·109-s + 4·113-s − 6·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 0.377·7-s − 2·9-s − 3.67·19-s − 6/5·25-s + 2.22·29-s − 2.87·31-s − 0.657·37-s + 2.33·47-s + 1/7·49-s + 1.64·53-s − 0.755·63-s + 3·81-s − 1.75·83-s + 3.15·103-s − 1.91·109-s + 0.376·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

Λ(s)=(43904s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(43904s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 4390443904    =    27⋅732^{7} \cdot 7^{3}
Sign: −1-1
Analytic conductor: 2.799352.79935
Root analytic conductor: 1.293491.29349
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 43904, ( :1/2,1/2), −1)(4,\ 43904,\ (\ :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 1 - T
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)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
31C2C_2 (1+8T+pT2)2 ( 1 + 8 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)(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)2 ( 1 - 6 T + p T^{2} )^{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)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
73C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
83C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
89C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 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

−10.17168816468660818702810083034, −9.117894034638208548514808203742, −8.772026867484659934825308819852, −8.560607806232820418165457845354, −8.078246359041317001571436271675, −7.30264187759595822059178656689, −6.67949194679031572918581545842, −5.92791093661634117809390012383, −5.82613210142107824134972732669, −4.98409920426632935260415692638, −4.18465932715971087267449515138, −3.68340574560716411313233602487, −2.50561452898491372410349134801, −2.14556791802447766709825303506, 0, 2.14556791802447766709825303506, 2.50561452898491372410349134801, 3.68340574560716411313233602487, 4.18465932715971087267449515138, 4.98409920426632935260415692638, 5.82613210142107824134972732669, 5.92791093661634117809390012383, 6.67949194679031572918581545842, 7.30264187759595822059178656689, 8.078246359041317001571436271675, 8.560607806232820418165457845354, 8.772026867484659934825308819852, 9.117894034638208548514808203742, 10.17168816468660818702810083034

Graph of the ZZ-function along the critical line