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Properties

Label 4-1216e2-1.1-c1e2-0-6
Degree 44
Conductor 14786561478656
Sign −1-1
Analytic cond. 94.280394.2803
Root an. cond. 3.116053.11605
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  − 6·5-s − 6·9-s + 8·13-s − 6·17-s + 17·25-s − 16·37-s + 36·45-s + 11·49-s + 12·53-s + 10·61-s − 48·65-s − 22·73-s + 27·81-s + 36·85-s − 20·89-s − 24·97-s + 4·101-s − 12·109-s − 8·113-s − 48·117-s + 3·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  − 2.68·5-s − 2·9-s + 2.21·13-s − 1.45·17-s + 17/5·25-s − 2.63·37-s + 5.36·45-s + 11/7·49-s + 1.64·53-s + 1.28·61-s − 5.95·65-s − 2.57·73-s + 3·81-s + 3.90·85-s − 2.11·89-s − 2.43·97-s + 0.398·101-s − 1.14·109-s − 0.752·113-s − 4.43·117-s + 3/11·121-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯

Functional equation

Λ(s)=(1478656s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1478656s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14786561478656    =    212⋅1922^{12} \cdot 19^{2}
Sign: −1-1
Analytic conductor: 94.280394.2803
Root analytic conductor: 3.116053.11605
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 1478656, ( :1/2,1/2), −1)(4,\ 1478656,\ (\ :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
19C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good3C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
5C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
7C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
11C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
13C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
17C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
31C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
37C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
47C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
61C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
67C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
71C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
73C2C_2 (1+11T+pT2)2 ( 1 + 11 T + p T^{2} )^{2}
79C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
97C2C_2 (1+12T+pT2)2 ( 1 + 12 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

−7.86378097245535978247876093758, −7.18059484484420096366454394254, −6.98286792123187236566096654142, −6.56370906167522668644909842214, −5.76283520417985127913399041730, −5.67103421999051188975454804232, −5.05210681704737714861605588166, −4.14385266867940367577820086585, −4.09350585892641493698723914863, −3.71418757784612877583046899350, −3.10630100563499367104455654686, −2.78320977751625677782435169813, −1.77861406991832868392707836598, −0.65261125406809044714433414915, 0, 0.65261125406809044714433414915, 1.77861406991832868392707836598, 2.78320977751625677782435169813, 3.10630100563499367104455654686, 3.71418757784612877583046899350, 4.09350585892641493698723914863, 4.14385266867940367577820086585, 5.05210681704737714861605588166, 5.67103421999051188975454804232, 5.76283520417985127913399041730, 6.56370906167522668644909842214, 6.98286792123187236566096654142, 7.18059484484420096366454394254, 7.86378097245535978247876093758

Graph of the ZZ-function along the critical line