Properties

Label 4-395136-1.1-c1e2-0-26
Degree 44
Conductor 395136395136
Sign 1-1
Analytic cond. 25.194225.1942
Root an. cond. 2.240392.24039
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 7-s − 8-s + 9-s − 4·11-s − 14-s + 16-s − 18-s + 4·22-s + 4·23-s − 2·25-s + 28-s + 4·29-s − 32-s + 36-s − 4·37-s − 12·43-s − 4·44-s − 4·46-s + 49-s + 2·50-s − 16·53-s − 56-s − 4·58-s + 63-s + 64-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.852·22-s + 0.834·23-s − 2/5·25-s + 0.188·28-s + 0.742·29-s − 0.176·32-s + 1/6·36-s − 0.657·37-s − 1.82·43-s − 0.603·44-s − 0.589·46-s + 1/7·49-s + 0.282·50-s − 2.19·53-s − 0.133·56-s − 0.525·58-s + 0.125·63-s + 1/8·64-s + ⋯

Functional equation

Λ(s)=(395136s/2ΓC(s)2L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(395136s/2ΓC(s+1/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 395136395136    =    2732732^{7} \cdot 3^{2} \cdot 7^{3}
Sign: 1-1
Analytic conductor: 25.194225.1942
Root analytic conductor: 2.240392.24039
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 395136, ( :1/2,1/2), 1)(4,\ 395136,\ (\ :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(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C1C_1 1+T 1 + T
3C1C_1×\timesC1C_1 (1T)(1+T) ( 1 - T )( 1 + T )
7C1C_1 1T 1 - T
good5C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
11C2C_2×\timesC2C_2 (1+pT2)(1+4T+pT2) ( 1 + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (14T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C22C_2^2 126T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
19C2C_2 (18T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
23C2C_2×\timesC2C_2 (18T+pT2)(1+4T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (12T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
31C22C_2^2 134T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
37C2C_2×\timesC2C_2 (12T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1+4T+pT2)(1+8T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
47C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
53C2C_2×\timesC2C_2 (1+2T+pT2)(1+14T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} )
59C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
61C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
67C2C_2 (112T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C2C_2×\timesC2C_2 (18T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
73C2C_2 (110T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C2C_2×\timesC2C_2 (1+pT2)(1+16T+pT2) ( 1 + p T^{2} )( 1 + 16 T + p T^{2} )
83C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
89C22C_2^2 1+30T2+p2T4 1 + 30 T^{2} + p^{2} T^{4}
97C22C_2^2 1+102T2+p2T4 1 + 102 T^{2} + p^{2} T^{4}
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   L(s)=p j=14(1αj,pps)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

−8.411521090255994624785953587423, −8.052477028057648756115602305972, −7.54628166700195695041542955996, −7.23098376625686350759207285220, −6.58757381477372958101054261593, −6.28921047893520197455827551076, −5.54286182184328238279245201745, −5.01012367862760280737746076876, −4.79245704236827232363207773370, −3.94307762210019179524961979588, −3.21664771874035875051559967518, −2.75886454611533471050190400956, −1.96516192096943474380630638016, −1.28353908819465834293012833127, 0, 1.28353908819465834293012833127, 1.96516192096943474380630638016, 2.75886454611533471050190400956, 3.21664771874035875051559967518, 3.94307762210019179524961979588, 4.79245704236827232363207773370, 5.01012367862760280737746076876, 5.54286182184328238279245201745, 6.28921047893520197455827551076, 6.58757381477372958101054261593, 7.23098376625686350759207285220, 7.54628166700195695041542955996, 8.052477028057648756115602305972, 8.411521090255994624785953587423

Graph of the ZZ-function along the critical line