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Properties

Label 8-637e4-1.1-c1e4-0-14
Degree 88
Conductor 164648481361164648481361
Sign 11
Analytic cond. 669.369669.369
Root an. cond. 2.255322.25532
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 3-s + 3·4-s + 3·6-s + 9-s − 9·11-s + 3·12-s − 14·13-s − 2·16-s + 6·17-s + 3·18-s + 9·19-s − 27·22-s − 6·23-s + 15·25-s − 42·26-s − 4·27-s + 9·29-s + 6·32-s − 9·33-s + 18·34-s + 3·36-s + 24·37-s + 27·38-s − 14·39-s − 18·41-s − 5·43-s + ⋯
L(s)  = 1  + 2.12·2-s + 0.577·3-s + 3/2·4-s + 1.22·6-s + 1/3·9-s − 2.71·11-s + 0.866·12-s − 3.88·13-s − 1/2·16-s + 1.45·17-s + 0.707·18-s + 2.06·19-s − 5.75·22-s − 1.25·23-s + 3·25-s − 8.23·26-s − 0.769·27-s + 1.67·29-s + 1.06·32-s − 1.56·33-s + 3.08·34-s + 1/2·36-s + 3.94·37-s + 4.37·38-s − 2.24·39-s − 2.81·41-s − 0.762·43-s + ⋯

Functional equation

Λ(s)=((78⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((78⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 78⋅1347^{8} \cdot 13^{4}
Sign: 11
Analytic conductor: 669.369669.369
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 78⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 7^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 6.3072197566.307219756
L(12)L(\frac12) ≈\approx 6.3072197566.307219756
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)
bad7 1 1
13C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
good2C2C_2×\timesC22C_2^2 (1−T+pT2)2(1−T−T2−pT3+p2T4) ( 1 - T + p T^{2} )^{2}( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} )
3D4×C2D_4\times C_2 1−T+5T3−11T4+5pT5−p3T7+p4T8 1 - T + 5 T^{3} - 11 T^{4} + 5 p T^{5} - p^{3} T^{7} + p^{4} T^{8}
5D4×C2D_4\times C_2 1−3pT2+101T4−3p3T6+p4T8 1 - 3 p T^{2} + 101 T^{4} - 3 p^{3} T^{6} + p^{4} T^{8}
11D4×C2D_4\times C_2 1+9T+54T2+243T3+905T4+243pT5+54p2T6+9p3T7+p4T8 1 + 9 T + 54 T^{2} + 243 T^{3} + 905 T^{4} + 243 p T^{5} + 54 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8}
17C22C_2^2 (1−3T−8T2−3pT3+p2T4)2 ( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}
19D4×C2D_4\times C_2 1−9T+56T2−261T3+993T4−261pT5+56p2T6−9p3T7+p4T8 1 - 9 T + 56 T^{2} - 261 T^{3} + 993 T^{4} - 261 p T^{5} + 56 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}
23D4×C2D_4\times C_2 1+6T+2T2−72T3−201T4−72pT5+2p2T6+6p3T7+p4T8 1 + 6 T + 2 T^{2} - 72 T^{3} - 201 T^{4} - 72 p T^{5} + 2 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
29D4×C2D_4\times C_2 1−9T+8T2−135T3+2139T4−135pT5+8p2T6−9p3T7+p4T8 1 - 9 T + 8 T^{2} - 135 T^{3} + 2139 T^{4} - 135 p T^{5} + 8 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}
31C2C_2 (1−7T+pT2)2(1+7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2}
37C2C_2 (1−11T+pT2)2(1−T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2}
41D4×C2D_4\times C_2 1+18T+210T2+1836T3+13151T4+1836pT5+210p2T6+18p3T7+p4T8 1 + 18 T + 210 T^{2} + 1836 T^{3} + 13151 T^{4} + 1836 p T^{5} + 210 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8}
43D4×C2D_4\times C_2 1+5T−20T2−205T3−899T4−205pT5−20p2T6+5p3T7+p4T8 1 + 5 T - 20 T^{2} - 205 T^{3} - 899 T^{4} - 205 p T^{5} - 20 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}
47D4×C2D_4\times C_2 1−78T2+4595T4−78p2T6+p4T8 1 - 78 T^{2} + 4595 T^{4} - 78 p^{2} T^{6} + p^{4} T^{8}
53D4D_{4} (1+6T+31T2+6pT3+p2T4)2 ( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
59D4×C2D_4\times C_2 1+6T+21T2+54T3−2692T4+54pT5+21p2T6+6p3T7+p4T8 1 + 6 T + 21 T^{2} + 54 T^{3} - 2692 T^{4} + 54 p T^{5} + 21 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
61D4×C2D_4\times C_2 1+2T+70T2−376T3+391T4−376pT5+70p2T6+2p3T7+p4T8 1 + 2 T + 70 T^{2} - 376 T^{3} + 391 T^{4} - 376 p T^{5} + 70 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
67C2C_2×\timesC22C_2^2 (1−4T+pT2)2(1−4T−51T2−4pT3+p2T4) ( 1 - 4 T + p T^{2} )^{2}( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )
71D4×C2D_4\times C_2 1−6T+150T2−828T3+14855T4−828pT5+150p2T6−6p3T7+p4T8 1 - 6 T + 150 T^{2} - 828 T^{3} + 14855 T^{4} - 828 p T^{5} + 150 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
73C22C_2^2 (1−134T2+p2T4)2 ( 1 - 134 T^{2} + p^{2} T^{4} )^{2}
79C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
83D4×C2D_4\times C_2 1−270T2+31667T4−270p2T6+p4T8 1 - 270 T^{2} + 31667 T^{4} - 270 p^{2} T^{6} + p^{4} T^{8}
89D4×C2D_4\times C_2 1−33T+588T2−7425T3+75011T4−7425pT5+588p2T6−33p3T7+p4T8 1 - 33 T + 588 T^{2} - 7425 T^{3} + 75011 T^{4} - 7425 p T^{5} + 588 p^{2} T^{6} - 33 p^{3} T^{7} + p^{4} T^{8}
97D4×C2D_4\times C_2 1−39T+812T2−11895T3+132795T4−11895pT5+812p2T6−39p3T7+p4T8 1 - 39 T + 812 T^{2} - 11895 T^{3} + 132795 T^{4} - 11895 p T^{5} + 812 p^{2} T^{6} - 39 p^{3} T^{7} + p^{4} T^{8}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.74877425168365594260237677889, −7.34150167552812145586146284221, −7.28464483994551031939350551727, −6.98189212705122811679862003759, −6.58528229148462483192431559472, −6.27429343856061941558013860965, −6.16473037672556940125161873214, −5.72595265434186874292787510418, −5.47607828098225386784987627837, −5.24256810482218738934865840794, −4.96206817363485237646196803907, −4.82805757096900479260107752463, −4.74073964159259720591452148693, −4.64977096967864172658224527500, −4.56667528476319610584813868021, −3.74252111441710993804958462428, −3.48497073732612541650986951648, −3.29498762492748357425125107314, −2.84146579155933846267424105971, −2.77811873909924684767672166968, −2.72149990093153078315780972755, −2.21130596328021549322311126868, −1.92137794749078930091982327511, −0.946768860521896181542105793764, −0.55432840540967627536145525108, 0.55432840540967627536145525108, 0.946768860521896181542105793764, 1.92137794749078930091982327511, 2.21130596328021549322311126868, 2.72149990093153078315780972755, 2.77811873909924684767672166968, 2.84146579155933846267424105971, 3.29498762492748357425125107314, 3.48497073732612541650986951648, 3.74252111441710993804958462428, 4.56667528476319610584813868021, 4.64977096967864172658224527500, 4.74073964159259720591452148693, 4.82805757096900479260107752463, 4.96206817363485237646196803907, 5.24256810482218738934865840794, 5.47607828098225386784987627837, 5.72595265434186874292787510418, 6.16473037672556940125161873214, 6.27429343856061941558013860965, 6.58528229148462483192431559472, 6.98189212705122811679862003759, 7.28464483994551031939350551727, 7.34150167552812145586146284221, 7.74877425168365594260237677889

Graph of the ZZ-function along the critical line