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Properties

Label 8-37e4-1.1-c1e4-0-0
Degree $8$
Conductor $1874161$
Sign $1$
Analytic cond. $0.00761930$
Root an. cond. $0.543549$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 3·4-s − 6·5-s + 4·9-s − 12·11-s − 6·12-s + 12·13-s − 12·15-s + 4·16-s + 6·17-s − 6·19-s + 18·20-s + 15·25-s + 4·27-s − 24·33-s − 12·36-s + 2·37-s + 24·39-s − 6·41-s + 36·44-s − 24·45-s + 12·47-s + 8·48-s + 2·49-s + 12·51-s − 36·52-s + 12·53-s + ⋯
L(s)  = 1  + 1.15·3-s − 3/2·4-s − 2.68·5-s + 4/3·9-s − 3.61·11-s − 1.73·12-s + 3.32·13-s − 3.09·15-s + 16-s + 1.45·17-s − 1.37·19-s + 4.02·20-s + 3·25-s + 0.769·27-s − 4.17·33-s − 2·36-s + 0.328·37-s + 3.84·39-s − 0.937·41-s + 5.42·44-s − 3.57·45-s + 1.75·47-s + 1.15·48-s + 2/7·49-s + 1.68·51-s − 4.99·52-s + 1.64·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1874161 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1874161 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(1874161\)    =    \(37^{4}\)
Sign: $1$
Analytic conductor: \(0.00761930\)
Root analytic conductor: \(0.543549\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 1874161,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2604158774\)
\(L(\frac12)\) \(\approx\) \(0.2604158774\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad37$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
good2$C_2^3$ \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \)
3$D_4\times C_2$ \( 1 - 2 T + 4 T^{3} - 5 T^{4} + 4 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2$$\times$$C_2^2$ \( ( 1 + 2 T + p T^{2} )^{2}( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
7$C_2^2$$\times$$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \)
11$D_{4}$ \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 6 T + 45 T^{2} - 198 T^{3} + 1004 T^{4} - 198 p T^{5} + 45 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 + 6 T + 44 T^{2} + 192 T^{3} + 891 T^{4} + 192 p T^{5} + 44 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 84 T^{2} + 2810 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 - 102 T^{2} + 4235 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 52 T^{2} + 1626 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
47$D_{4}$ \( ( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 12 T + 14 T^{2} - 288 T^{3} + 6459 T^{4} - 288 p T^{5} + 14 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 12 T + 78 T^{2} + 360 T^{3} + 251 T^{4} + 360 p T^{5} + 78 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 6 T + 101 T^{2} + 534 T^{3} + 4932 T^{4} + 534 p T^{5} + 101 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 18 T + 112 T^{2} - 1404 T^{3} + 18747 T^{4} - 1404 p T^{5} + 112 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + p T^{2} )^{4} \)
79$D_4\times C_2$ \( 1 + 18 T + 212 T^{2} + 1872 T^{3} + 13107 T^{4} + 1872 p T^{5} + 212 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 6 T - 112 T^{2} + 108 T^{3} + 12027 T^{4} + 108 p T^{5} - 112 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 18 T + 297 T^{2} - 3402 T^{3} + 37412 T^{4} - 3402 p T^{5} + 297 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 310 T^{2} + 42411 T^{4} - 310 p^{2} T^{6} + p^{4} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.82142626557256032763234170957, −12.04491838717221638246372863411, −11.83719433424265678692159522193, −11.56789388667199248067801007828, −10.79307589990860853553547145219, −10.66542876398739700939841692566, −10.61099437881020159914628383184, −10.34099957354795295676075951142, −9.667180646149637157853060340803, −9.266411750997812188935422250362, −8.615746368426324840805977490267, −8.503065513922239613514066855566, −8.463055515120429309058165103598, −7.981710621679176466758122142059, −7.67398301586931020584067523025, −7.63988918284055115668979239788, −7.11439599612220304666639934573, −6.21368134975019881540540966060, −5.71895029011267292383692650485, −5.27240718439870873687442968740, −4.63135584885078132368899514754, −4.13590772308870469558933499340, −3.85722758301771213994367792375, −3.47142450387139421332160195020, −2.83419209981600773873845599856, 2.83419209981600773873845599856, 3.47142450387139421332160195020, 3.85722758301771213994367792375, 4.13590772308870469558933499340, 4.63135584885078132368899514754, 5.27240718439870873687442968740, 5.71895029011267292383692650485, 6.21368134975019881540540966060, 7.11439599612220304666639934573, 7.63988918284055115668979239788, 7.67398301586931020584067523025, 7.981710621679176466758122142059, 8.463055515120429309058165103598, 8.503065513922239613514066855566, 8.615746368426324840805977490267, 9.266411750997812188935422250362, 9.667180646149637157853060340803, 10.34099957354795295676075951142, 10.61099437881020159914628383184, 10.66542876398739700939841692566, 10.79307589990860853553547145219, 11.56789388667199248067801007828, 11.83719433424265678692159522193, 12.04491838717221638246372863411, 12.82142626557256032763234170957

Graph of the $Z$-function along the critical line