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Properties

Label 4-320e2-1.1-c1e2-0-17
Degree 44
Conductor 102400102400
Sign 11
Analytic cond. 6.529116.52911
Root an. cond. 1.598501.59850
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 − 2·5-s + 2·7-s + 2·9-s + 8·11-s − 6·13-s − 4·15-s − 6·17-s + 4·21-s + 6·23-s − 25-s + 6·27-s + 4·29-s + 16·33-s − 4·35-s − 6·37-s − 12·39-s + 12·41-s − 6·43-s − 4·45-s + 18·47-s + 2·49-s − 12·51-s + 10·53-s − 16·55-s + 4·63-s + 12·65-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 0.755·7-s + 2/3·9-s + 2.41·11-s − 1.66·13-s − 1.03·15-s − 1.45·17-s + 0.872·21-s + 1.25·23-s − 1/5·25-s + 1.15·27-s + 0.742·29-s + 2.78·33-s − 0.676·35-s − 0.986·37-s − 1.92·39-s + 1.87·41-s − 0.914·43-s − 0.596·45-s + 2.62·47-s + 2/7·49-s − 1.68·51-s + 1.37·53-s − 2.15·55-s + 0.503·63-s + 1.48·65-s + ⋯

Functional equation

Λ(s)=(102400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(102400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 102400102400    =    212⋅522^{12} \cdot 5^{2}
Sign: 11
Analytic conductor: 6.529116.52911
Root analytic conductor: 1.598501.59850
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 102400, ( :1/2,1/2), 1)(4,\ 102400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2521343342.252134334
L(12)L(\frac12) ≈\approx 2.2521343342.252134334
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
5C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
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}
11C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
13C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
17C2C_2 (1−2T+pT2)(1+8T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
23C22C_2^2 1−6T+18T2−6pT3+p2T4 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}
29C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
31C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
37C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
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 1−18T+162T2−18pT3+p2T4 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4}
53C2C_2 (1−14T+pT2)(1+4T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} )
59C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
61C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
67C22C_2^2 1+18T+162T2+18pT3+p2T4 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−106T2+p2T4 1 - 106 T^{2} + p^{2} T^{4}
73C2C_2 (1−16T+pT2)(1+6T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
89C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
97C22C_2^2 1+14T+98T2+14pT3+p2T4 1 + 14 T + 98 T^{2} + 14 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

−11.81113829924476911159363322084, −11.61430812631216105871510866690, −10.90084021176686885646320573268, −10.59988447635737109397568659638, −9.872085441494781428469158973204, −9.173714116762044901093328545790, −9.080884202459432893602604602317, −8.699215331916632323856246555226, −8.194058849569125180362096735563, −7.52852954767173224025226745630, −7.00410605261467062306638326401, −6.96678563256221610647661711931, −6.14282944022681839691336982259, −5.21359479119464999923726927145, −4.38548362701850026288655848623, −4.36018674895597463202904052663, −3.67802525611135460159399783198, −2.78602271721816750954710911455, −2.23191198852958141835641901402, −1.14701040917596716806492373676, 1.14701040917596716806492373676, 2.23191198852958141835641901402, 2.78602271721816750954710911455, 3.67802525611135460159399783198, 4.36018674895597463202904052663, 4.38548362701850026288655848623, 5.21359479119464999923726927145, 6.14282944022681839691336982259, 6.96678563256221610647661711931, 7.00410605261467062306638326401, 7.52852954767173224025226745630, 8.194058849569125180362096735563, 8.699215331916632323856246555226, 9.080884202459432893602604602317, 9.173714116762044901093328545790, 9.872085441494781428469158973204, 10.59988447635737109397568659638, 10.90084021176686885646320573268, 11.61430812631216105871510866690, 11.81113829924476911159363322084

Graph of the ZZ-function along the critical line