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Properties

Label 4-442368-1.1-c1e2-0-29
Degree 44
Conductor 442368442368
Sign −1-1
Analytic cond. 28.205728.2057
Root an. cond. 2.304542.30454
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 8·11-s + 6·25-s − 27-s + 8·33-s + 14·49-s + 8·59-s − 20·73-s − 6·75-s + 81-s + 8·83-s + 4·97-s − 8·99-s − 24·107-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 14·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 2.41·11-s + 6/5·25-s − 0.192·27-s + 1.39·33-s + 2·49-s + 1.04·59-s − 2.34·73-s − 0.692·75-s + 1/9·81-s + 0.878·83-s + 0.406·97-s − 0.804·99-s − 2.32·107-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.15·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + ⋯

Functional equation

Λ(s)=(442368s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(442368s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 442368442368    =    214⋅332^{14} \cdot 3^{3}
Sign: −1-1
Analytic conductor: 28.205728.2057
Root analytic conductor: 2.304542.30454
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 442368, ( :1/2,1/2), −1)(4,\ 442368,\ (\ :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
3C1C_1 1+T 1 + T
good5C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
7C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
11C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{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)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
29C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
31C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
37C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{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)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C22C_2^2 1−102T2+p2T4 1 - 102 T^{2} + p^{2} T^{4}
59C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
61C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 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)2 ( 1 + 10 T + p T^{2} )^{2}
79C22C_2^2 1−94T2+p2T4 1 - 94 T^{2} + p^{2} T^{4}
83C2C_2 (1−4T+pT2)2 ( 1 - 4 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 (1−2T+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

−8.236445515598692477888723219975, −7.926610475621468474965221388994, −7.39155350910112571498516060767, −7.08099310272590020465162708141, −6.54359139482422436566039922992, −5.85823971080050356977456524365, −5.48995581384585090744623344678, −5.15608716920273791676638800392, −4.63446424071817154248333823949, −4.09124569796799304831854781613, −3.27547083513429128978919773001, −2.66556849206542542189404706592, −2.25484300424030316326061471423, −1.08342707867212411461776876386, 0, 1.08342707867212411461776876386, 2.25484300424030316326061471423, 2.66556849206542542189404706592, 3.27547083513429128978919773001, 4.09124569796799304831854781613, 4.63446424071817154248333823949, 5.15608716920273791676638800392, 5.48995581384585090744623344678, 5.85823971080050356977456524365, 6.54359139482422436566039922992, 7.08099310272590020465162708141, 7.39155350910112571498516060767, 7.926610475621468474965221388994, 8.236445515598692477888723219975

Graph of the ZZ-function along the critical line