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Properties

Label 4-819e2-1.1-c1e2-0-18
Degree 44
Conductor 670761670761
Sign 11
Analytic cond. 42.768342.7683
Root an. cond. 2.557292.55729
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  − 2·2-s − 4-s + 3·5-s + 4·7-s + 8·8-s − 6·10-s − 3·11-s − 2·13-s − 8·14-s − 7·16-s + 4·17-s + 19-s − 3·20-s + 6·22-s + 5·25-s + 4·26-s − 4·28-s + 7·29-s − 3·31-s − 14·32-s − 8·34-s + 12·35-s + 4·37-s − 2·38-s + 24·40-s + 3·41-s + 7·43-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s + 1.34·5-s + 1.51·7-s + 2.82·8-s − 1.89·10-s − 0.904·11-s − 0.554·13-s − 2.13·14-s − 7/4·16-s + 0.970·17-s + 0.229·19-s − 0.670·20-s + 1.27·22-s + 25-s + 0.784·26-s − 0.755·28-s + 1.29·29-s − 0.538·31-s − 2.47·32-s − 1.37·34-s + 2.02·35-s + 0.657·37-s − 0.324·38-s + 3.79·40-s + 0.468·41-s + 1.06·43-s + ⋯

Functional equation

Λ(s)=(670761s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 670761 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(670761s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 670761 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 670761670761    =    34⋅72⋅1323^{4} \cdot 7^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 42.768342.7683
Root analytic conductor: 2.557292.55729
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 670761, ( :1/2,1/2), 1)(4,\ 670761,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1749172831.174917283
L(12)L(\frac12) ≈\approx 1.1749172831.174917283
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)
bad3 1 1
7C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
13C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
good2C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
5C22C_2^2 1−3T+4T2−3pT3+p2T4 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4}
11C22C_2^2 1+3T−2T2+3pT3+p2T4 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4}
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19C2C_2 (1−8T+pT2)(1+7T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} )
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C22C_2^2 1−7T+20T2−7pT3+p2T4 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4}
31C22C_2^2 1+3T−22T2+3pT3+p2T4 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4}
37C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
41C22C_2^2 1−3T−32T2−3pT3+p2T4 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4}
43C22C_2^2 1−7T+6T2−7pT3+p2T4 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−T−46T2−pT3+p2T4 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4}
53C22C_2^2 1−3T−44T2−3pT3+p2T4 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4}
59C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
61C2C_2 (1−14T+pT2)(1+T+pT2) ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} )
67C22C_2^2 1−3T−58T2−3pT3+p2T4 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−13T+98T2−13pT3+p2T4 1 - 13 T + 98 T^{2} - 13 p T^{3} + p^{2} T^{4}
73C22C_2^2 1−13T+96T2−13pT3+p2T4 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4}
79C22C_2^2 1−3T−70T2−3pT3+p2T4 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C2C_2 (1−19T+pT2)(1+14T+pT2) ( 1 - 19 T + p T^{2} )( 1 + 14 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

−10.19498515193657594743587468463, −9.939165597226645645218230901820, −9.561407804816850601755642414400, −9.343161768955426914645143453045, −8.746883799490829594903641028401, −8.300126757796985391135939740893, −8.044422464239794806373274107640, −7.85170734237166521008911248876, −7.10504011516982068271075081864, −6.94443125250656373439691471542, −5.79822073224444378540588248686, −5.58625417441522171535448808730, −4.99660545715066328729164023005, −4.90940484260439897130647317915, −4.23956269856436241918243244399, −3.62128743612012108477951849056, −2.43760343044770223921198018033, −2.18038063981077940198230753869, −1.23821931987795033679240901262, −0.835655756923601098163415042913, 0.835655756923601098163415042913, 1.23821931987795033679240901262, 2.18038063981077940198230753869, 2.43760343044770223921198018033, 3.62128743612012108477951849056, 4.23956269856436241918243244399, 4.90940484260439897130647317915, 4.99660545715066328729164023005, 5.58625417441522171535448808730, 5.79822073224444378540588248686, 6.94443125250656373439691471542, 7.10504011516982068271075081864, 7.85170734237166521008911248876, 8.044422464239794806373274107640, 8.300126757796985391135939740893, 8.746883799490829594903641028401, 9.343161768955426914645143453045, 9.561407804816850601755642414400, 9.939165597226645645218230901820, 10.19498515193657594743587468463

Graph of the ZZ-function along the critical line