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Properties

Label 4-640e2-1.1-c1e2-0-4
Degree 44
Conductor 409600409600
Sign 11
Analytic cond. 26.116426.1164
Root an. cond. 2.260622.26062
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 + 2·9-s − 6·11-s − 6·13-s − 4·15-s − 2·19-s + 16·23-s − 25-s + 6·27-s − 6·29-s − 12·33-s − 6·37-s − 12·39-s + 6·43-s − 4·45-s − 14·49-s + 18·53-s + 12·55-s − 4·57-s + 18·59-s + 10·61-s + 12·65-s − 6·67-s + 32·69-s + 12·73-s − 2·75-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 2/3·9-s − 1.80·11-s − 1.66·13-s − 1.03·15-s − 0.458·19-s + 3.33·23-s − 1/5·25-s + 1.15·27-s − 1.11·29-s − 2.08·33-s − 0.986·37-s − 1.92·39-s + 0.914·43-s − 0.596·45-s − 2·49-s + 2.47·53-s + 1.61·55-s − 0.529·57-s + 2.34·59-s + 1.28·61-s + 1.48·65-s − 0.733·67-s + 3.85·69-s + 1.40·73-s − 0.230·75-s + ⋯

Functional equation

Λ(s)=(409600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(409600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 409600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 409600409600    =    214⋅522^{14} \cdot 5^{2}
Sign: 11
Analytic conductor: 26.116426.1164
Root analytic conductor: 2.260622.26062
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 409600, ( :1/2,1/2), 1)(4,\ 409600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5813149621.581314962
L(12)L(\frac12) ≈\approx 1.5813149621.581314962
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}
7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
13C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
17C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
19C22C_2^2 1+2T+2T2+2pT3+p2T4 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
23C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
29C2C_2 (1−4T+pT2)(1+10T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} )
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
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−pT2)2 ( 1 - 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 1−90T2+p2T4 1 - 90 T^{2} + 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 1−18T+162T2−18pT3+p2T4 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−10T+50T2−10pT3+p2T4 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−106T2+p2T4 1 - 106 T^{2} + p^{2} T^{4}
73C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
79C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
83C22C_2^2 1−18T+162T2−18pT3+p2T4 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4}
89C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
97C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + 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

−10.74882932097001688569872328489, −10.24428104584806873137314125239, −10.02422947359015508493052249545, −9.364282824778839666626575241565, −8.931900027358690525427905030296, −8.696290961091251964030592508103, −7.967965778865657239327681287075, −7.934188730423437248325410580756, −7.28853589815066075941469339499, −6.99797587148800500169818590833, −6.72687674706282178522360407733, −5.43700459579989669504937424426, −5.24512710445198219723813836794, −4.90268197831600952299765836974, −4.16645413286408649562553726047, −3.60532436475525668714739903053, −2.78832385338468890164191577751, −2.78011231566836806873499945014, −2.03699737882536283037917391263, −0.62029260297903430405597842059, 0.62029260297903430405597842059, 2.03699737882536283037917391263, 2.78011231566836806873499945014, 2.78832385338468890164191577751, 3.60532436475525668714739903053, 4.16645413286408649562553726047, 4.90268197831600952299765836974, 5.24512710445198219723813836794, 5.43700459579989669504937424426, 6.72687674706282178522360407733, 6.99797587148800500169818590833, 7.28853589815066075941469339499, 7.934188730423437248325410580756, 7.967965778865657239327681287075, 8.696290961091251964030592508103, 8.931900027358690525427905030296, 9.364282824778839666626575241565, 10.02422947359015508493052249545, 10.24428104584806873137314125239, 10.74882932097001688569872328489

Graph of the ZZ-function along the critical line