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Properties

Label 4-3192e2-1.1-c1e2-0-3
Degree 44
Conductor 1018886410188864
Sign 11
Analytic cond. 649.650649.650
Root an. cond. 5.048585.04858
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·5-s + 2·7-s + 3·9-s − 8·11-s + 4·13-s − 4·15-s − 10·17-s + 2·19-s + 4·21-s − 4·25-s + 4·27-s − 10·29-s − 4·31-s − 16·33-s − 4·35-s + 4·37-s + 8·39-s − 4·41-s − 4·43-s − 6·45-s − 10·47-s + 3·49-s − 20·51-s − 2·53-s + 16·55-s + 4·57-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 0.755·7-s + 9-s − 2.41·11-s + 1.10·13-s − 1.03·15-s − 2.42·17-s + 0.458·19-s + 0.872·21-s − 4/5·25-s + 0.769·27-s − 1.85·29-s − 0.718·31-s − 2.78·33-s − 0.676·35-s + 0.657·37-s + 1.28·39-s − 0.624·41-s − 0.609·43-s − 0.894·45-s − 1.45·47-s + 3/7·49-s − 2.80·51-s − 0.274·53-s + 2.15·55-s + 0.529·57-s + ⋯

Functional equation

Λ(s)=(10188864s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 10188864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(10188864s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10188864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1018886410188864    =    26⋅32⋅72⋅1922^{6} \cdot 3^{2} \cdot 7^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 649.650649.650
Root analytic conductor: 5.048585.04858
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 10188864, ( :1/2,1/2), 1)(4,\ 10188864,\ (\ :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)2 ( 1 - T )^{2}
7C1C_1 (1−T)2 ( 1 - T )^{2}
19C1C_1 (1−T)2 ( 1 - T )^{2}
good5D4D_{4} 1+2T+8T2+2pT3+p2T4 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4}
11C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
13D4D_{4} 1−4T+18T2−4pT3+p2T4 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4}
17D4D_{4} 1+10T+56T2+10pT3+p2T4 1 + 10 T + 56 T^{2} + 10 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
29D4D_{4} 1+10T+80T2+10pT3+p2T4 1 + 10 T + 80 T^{2} + 10 p T^{3} + p^{2} T^{4}
31C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
37D4D_{4} 1−4T+30T2−4pT3+p2T4 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+4T+38T2+4pT3+p2T4 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+4T+42T2+4pT3+p2T4 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+10T+92T2+10pT3+p2T4 1 + 10 T + 92 T^{2} + 10 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+2T+32T2+2pT3+p2T4 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+8T+86T2+8pT3+p2T4 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+12T+110T2+12pT3+p2T4 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−4T+126T2−4pT3+p2T4 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+2T+116T2+2pT3+p2T4 1 + 2 T + 116 T^{2} + 2 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
79D4D_{4} 1+8T+126T2+8pT3+p2T4 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−6T+148T2−6pT3+p2T4 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97D4D_{4} 1+4T+6T2+4pT3+p2T4 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4}
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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

−8.233094778261918892145358667709, −8.215080694543461831295586226946, −7.73591949671411004845472382152, −7.69077403006010244619603218209, −7.01509303503386221066223596200, −6.96597955757863350195984661075, −6.10424659975622265851487578233, −5.91836790843344527318839669470, −5.24732949074927538314814056862, −4.99746464758065943525636689581, −4.38183196384062589024908734169, −4.32522413119396594845933725526, −3.51066016760584286453364321721, −3.48395451475226006682558209041, −2.74266292618532147556381359504, −2.38738853561610578284772266510, −1.83437597084645015196398689254, −1.50974378961662278794505562502, 0, 0, 1.50974378961662278794505562502, 1.83437597084645015196398689254, 2.38738853561610578284772266510, 2.74266292618532147556381359504, 3.48395451475226006682558209041, 3.51066016760584286453364321721, 4.32522413119396594845933725526, 4.38183196384062589024908734169, 4.99746464758065943525636689581, 5.24732949074927538314814056862, 5.91836790843344527318839669470, 6.10424659975622265851487578233, 6.96597955757863350195984661075, 7.01509303503386221066223596200, 7.69077403006010244619603218209, 7.73591949671411004845472382152, 8.215080694543461831295586226946, 8.233094778261918892145358667709

Graph of the ZZ-function along the critical line