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Properties

Label 4-40e4-1.1-c1e2-0-22
Degree 44
Conductor 25600002560000
Sign 11
Analytic cond. 163.227163.227
Root an. cond. 3.574363.57436
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·7-s + 2·9-s − 2·13-s − 2·17-s + 8·19-s − 4·21-s + 10·23-s + 6·27-s − 10·37-s − 4·39-s + 12·41-s − 6·43-s + 14·47-s + 2·49-s − 4·51-s − 2·53-s + 16·57-s + 8·59-s − 4·61-s − 4·63-s + 14·67-s + 20·69-s − 18·73-s + 16·79-s + 11·81-s + 10·83-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s + 2/3·9-s − 0.554·13-s − 0.485·17-s + 1.83·19-s − 0.872·21-s + 2.08·23-s + 1.15·27-s − 1.64·37-s − 0.640·39-s + 1.87·41-s − 0.914·43-s + 2.04·47-s + 2/7·49-s − 0.560·51-s − 0.274·53-s + 2.11·57-s + 1.04·59-s − 0.512·61-s − 0.503·63-s + 1.71·67-s + 2.40·69-s − 2.10·73-s + 1.80·79-s + 11/9·81-s + 1.09·83-s + ⋯

Functional equation

Λ(s)=(2560000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2560000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 25600002560000    =    212⋅542^{12} \cdot 5^{4}
Sign: 11
Analytic conductor: 163.227163.227
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2560000, ( :1/2,1/2), 1)(4,\ 2560000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.5417918273.541791827
L(12)L(\frac12) ≈\approx 3.5417918273.541791827
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
5 1 1
good3C22C_2^2 1−2T+2T2−2pT3+p2T4 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}
7C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
11C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
13C2C_2 (1−4T+pT2)(1+6T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C22C_2^2 1−10T+50T2−10pT3+p2T4 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
31C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
37C2C_2 (1−2T+pT2)(1+12T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−14T+98T2−14pT3+p2T4 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4}
53C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
59C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
61C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
67C22C_2^2 1−14T+98T2−14pT3+p2T4 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−106T2+p2T4 1 - 106 T^{2} + p^{2} T^{4}
73C22C_2^2 1+18T+162T2+18pT3+p2T4 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C22C_2^2 1−10T+50T2−10pT3+p2T4 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4}
89C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
97C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 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

−9.399138532315601929631861252584, −9.243072371392499420253651797293, −8.797798566465001675462334830021, −8.683378625186750792913895541591, −7.908526189653864853575777906300, −7.69670304685218610623931435606, −7.16307409618371448895878095344, −6.93727411485892768383615336976, −6.62869100103402877147323772212, −5.95393472857215121955830003603, −5.33786599831243196001657380320, −5.15604000074851863198251272913, −4.59793702833499293242457194665, −4.01837887188466131351765351014, −3.34403631815961901670637454084, −3.28253317423061125992855079629, −2.55911487843505104698930491476, −2.39312566231043210505475636148, −1.33689939001090117967721838301, −0.73496640570089924081057845686, 0.73496640570089924081057845686, 1.33689939001090117967721838301, 2.39312566231043210505475636148, 2.55911487843505104698930491476, 3.28253317423061125992855079629, 3.34403631815961901670637454084, 4.01837887188466131351765351014, 4.59793702833499293242457194665, 5.15604000074851863198251272913, 5.33786599831243196001657380320, 5.95393472857215121955830003603, 6.62869100103402877147323772212, 6.93727411485892768383615336976, 7.16307409618371448895878095344, 7.69670304685218610623931435606, 7.908526189653864853575777906300, 8.683378625186750792913895541591, 8.797798566465001675462334830021, 9.243072371392499420253651797293, 9.399138532315601929631861252584

Graph of the ZZ-function along the critical line