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Properties

Label 4-444528-1.1-c1e2-0-20
Degree 44
Conductor 444528444528
Sign −1-1
Analytic cond. 28.343428.3434
Root an. cond. 2.307342.30734
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  + 2-s − 4-s + 7-s − 3·8-s + 14-s − 16-s − 8·19-s − 6·25-s − 28-s + 4·29-s + 5·32-s + 12·37-s − 8·38-s + 49-s − 6·50-s − 12·53-s − 3·56-s + 4·58-s + 24·59-s + 7·64-s + 12·74-s + 8·76-s − 24·83-s + 98-s + 6·100-s − 16·103-s − 12·106-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s + 0.267·14-s − 1/4·16-s − 1.83·19-s − 6/5·25-s − 0.188·28-s + 0.742·29-s + 0.883·32-s + 1.97·37-s − 1.29·38-s + 1/7·49-s − 0.848·50-s − 1.64·53-s − 0.400·56-s + 0.525·58-s + 3.12·59-s + 7/8·64-s + 1.39·74-s + 0.917·76-s − 2.63·83-s + 0.101·98-s + 3/5·100-s − 1.57·103-s − 1.16·106-s + ⋯

Functional equation

Λ(s)=(444528s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 444528 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(444528s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 444528 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 444528444528    =    24⋅34⋅732^{4} \cdot 3^{4} \cdot 7^{3}
Sign: −1-1
Analytic conductor: 28.343428.3434
Root analytic conductor: 2.307342.30734
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 444528, ( :1/2,1/2), −1)(4,\ 444528,\ (\ :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)
bad2C2C_2 1−T+pT2 1 - T + p T^{2}
3 1 1
7C1C_1 1−T 1 - T
good5C2C_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+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{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 (1−2T+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+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
59C2C_2 (1−12T+pT2)2 ( 1 - 12 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+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 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+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
89C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
97C2C_2 (1−18T+pT2)(1+18T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 18 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

−8.351153010775658504468507691128, −8.123916451859490481483222873179, −7.45452630662509700459332399892, −6.81422856027217448872168795499, −6.43539728173776997986575871832, −5.76945209288606009941542010236, −5.73267711633132091527857634664, −4.75834670654208540745088589071, −4.61914309368534933978095799917, −3.94328917656394473634657471263, −3.71279869000131289976682575519, −2.64839746138688614832708523229, −2.38203049551357769485459868076, −1.27003349896142462152543818564, 0, 1.27003349896142462152543818564, 2.38203049551357769485459868076, 2.64839746138688614832708523229, 3.71279869000131289976682575519, 3.94328917656394473634657471263, 4.61914309368534933978095799917, 4.75834670654208540745088589071, 5.73267711633132091527857634664, 5.76945209288606009941542010236, 6.43539728173776997986575871832, 6.81422856027217448872168795499, 7.45452630662509700459332399892, 8.123916451859490481483222873179, 8.351153010775658504468507691128

Graph of the ZZ-function along the critical line