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Properties

Label 8-75e4-1.1-c4e4-0-1
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $3612.62$
Root an. cond. $2.78437$
Motivic weight $4$
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  − 12·2-s + 72·4-s + 72·7-s − 264·8-s − 24·11-s − 144·13-s − 864·14-s + 220·16-s − 600·17-s + 288·22-s − 888·23-s + 1.72e3·26-s + 5.18e3·28-s + 3.24e3·31-s + 4.32e3·32-s + 7.20e3·34-s − 2.44e3·37-s + 6.86e3·41-s + 5.54e3·43-s − 1.72e3·44-s + 1.06e4·46-s − 5.61e3·47-s + 2.59e3·49-s − 1.03e4·52-s − 1.84e3·53-s − 1.90e4·56-s + 1.50e4·61-s + ⋯
L(s)  = 1  − 3·2-s + 9/2·4-s + 1.46·7-s − 4.12·8-s − 0.198·11-s − 0.852·13-s − 4.40·14-s + 0.859·16-s − 2.07·17-s + 0.595·22-s − 1.67·23-s + 2.55·26-s + 6.61·28-s + 3.37·31-s + 4.21·32-s + 6.22·34-s − 1.78·37-s + 4.08·41-s + 2.99·43-s − 0.892·44-s + 5.03·46-s − 2.54·47-s + 1.07·49-s − 3.83·52-s − 0.657·53-s − 6.06·56-s + 4.03·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(3612.62\)
Root analytic conductor: \(2.78437\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :2, 2, 2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.01002621387\)
\(L(\frac12)\) \(\approx\) \(0.01002621387\)
\(L(3)\) 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)$
bad3$C_2^2$ \( 1 + p^{6} T^{4} \)
5 \( 1 \)
good2$D_4\times C_2$ \( 1 + 3 p^{2} T + 9 p^{3} T^{2} + 33 p^{3} T^{3} + 233 p^{2} T^{4} + 33 p^{7} T^{5} + 9 p^{11} T^{6} + 3 p^{14} T^{7} + p^{16} T^{8} \)
7$D_4\times C_2$ \( 1 - 72 T + 2592 T^{2} - 219312 T^{3} + 18140207 T^{4} - 219312 p^{4} T^{5} + 2592 p^{8} T^{6} - 72 p^{12} T^{7} + p^{16} T^{8} \)
11$D_{4}$ \( ( 1 + 12 T - 7186 T^{2} + 12 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + 144 T + 10368 T^{2} + 2429280 T^{3} + 432514319 T^{4} + 2429280 p^{4} T^{5} + 10368 p^{8} T^{6} + 144 p^{12} T^{7} + p^{16} T^{8} \)
17$D_4\times C_2$ \( 1 + 600 T + 180000 T^{2} + 77105400 T^{3} + 31005206018 T^{4} + 77105400 p^{4} T^{5} + 180000 p^{8} T^{6} + 600 p^{12} T^{7} + p^{16} T^{8} \)
19$D_4\times C_2$ \( 1 - 326690 T^{2} + 54622417923 T^{4} - 326690 p^{8} T^{6} + p^{16} T^{8} \)
23$D_4\times C_2$ \( 1 + 888 T + 394272 T^{2} + 52907928 T^{3} - 41414676478 T^{4} + 52907928 p^{4} T^{5} + 394272 p^{8} T^{6} + 888 p^{12} T^{7} + p^{16} T^{8} \)
29$D_4\times C_2$ \( 1 - 63932 p T^{2} + 1839990957414 T^{4} - 63932 p^{9} T^{6} + p^{16} T^{8} \)
31$D_{4}$ \( ( 1 - 1622 T + 2141667 T^{2} - 1622 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 2448 T + 2996352 T^{2} + 6308792208 T^{3} + 12788952535682 T^{4} + 6308792208 p^{4} T^{5} + 2996352 p^{8} T^{6} + 2448 p^{12} T^{7} + p^{16} T^{8} \)
41$D_{4}$ \( ( 1 - 3432 T + 8077562 T^{2} - 3432 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 5544 T + 15367968 T^{2} - 34365692592 T^{3} + 69120271160159 T^{4} - 34365692592 p^{4} T^{5} + 15367968 p^{8} T^{6} - 5544 p^{12} T^{7} + p^{16} T^{8} \)
47$D_4\times C_2$ \( 1 + 5616 T + 15769728 T^{2} + 24055647408 T^{3} + 36339718329314 T^{4} + 24055647408 p^{4} T^{5} + 15769728 p^{8} T^{6} + 5616 p^{12} T^{7} + p^{16} T^{8} \)
53$D_4\times C_2$ \( 1 + 1848 T + 1707552 T^{2} + 12837022968 T^{3} + 95614873611362 T^{4} + 12837022968 p^{4} T^{5} + 1707552 p^{8} T^{6} + 1848 p^{12} T^{7} + p^{16} T^{8} \)
59$D_4\times C_2$ \( 1 - 10941820 T^{2} + 100216939084998 T^{4} - 10941820 p^{8} T^{6} + p^{16} T^{8} \)
61$D_{4}$ \( ( 1 - 7510 T + 35401563 T^{2} - 7510 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 13320 T + 88711200 T^{2} - 439054559280 T^{3} + 2008874003467343 T^{4} - 439054559280 p^{4} T^{5} + 88711200 p^{8} T^{6} - 13320 p^{12} T^{7} + p^{16} T^{8} \)
71$D_{4}$ \( ( 1 + 10776 T + 74664506 T^{2} + 10776 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 19728 T + 194596992 T^{2} + 1434198789648 T^{3} + 8607659366333762 T^{4} + 1434198789648 p^{4} T^{5} + 194596992 p^{8} T^{6} + 19728 p^{12} T^{7} + p^{16} T^{8} \)
79$D_4\times C_2$ \( 1 - 25715324 T^{2} + 2139409705359366 T^{4} - 25715324 p^{8} T^{6} + p^{16} T^{8} \)
83$D_4\times C_2$ \( 1 + 8592 T + 36911232 T^{2} + 464368361424 T^{3} + 5798663913908642 T^{4} + 464368361424 p^{4} T^{5} + 36911232 p^{8} T^{6} + 8592 p^{12} T^{7} + p^{16} T^{8} \)
89$D_4\times C_2$ \( 1 - 203730340 T^{2} + 18051837318885318 T^{4} - 203730340 p^{8} T^{6} + p^{16} T^{8} \)
97$D_4\times C_2$ \( 1 - 11520 T + 66355200 T^{2} - 1104271879680 T^{3} + 18323409093272303 T^{4} - 1104271879680 p^{4} T^{5} + 66355200 p^{8} T^{6} - 11520 p^{12} T^{7} + p^{16} T^{8} \)
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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

−9.964095571279687791110569319213, −9.963628164375895277529852286287, −9.190149984781258471495861794573, −8.961837451665461991740794590060, −8.934797914532289439502647803643, −8.537336106092434049922447047236, −8.230277897450109010067023540050, −8.047146849977746708416450894584, −7.78019010227179610867979349942, −7.24287805417863313466005197587, −7.21844494507792221908312935788, −6.85115243911160431006956564254, −6.21878261684051567398131830055, −6.10991189879149770116406898852, −5.61187047689635363766717878589, −4.77856881431969079922560345576, −4.51240597080910658627572458635, −4.48473388293431638308550913542, −3.91748909226179845771219584062, −2.55972752525827784215757698008, −2.37989792492852585213167979234, −2.29704138032690019259725319316, −1.31101374695103413165768420687, −0.999655996067606689903239902481, −0.05064197751212270300992743704, 0.05064197751212270300992743704, 0.999655996067606689903239902481, 1.31101374695103413165768420687, 2.29704138032690019259725319316, 2.37989792492852585213167979234, 2.55972752525827784215757698008, 3.91748909226179845771219584062, 4.48473388293431638308550913542, 4.51240597080910658627572458635, 4.77856881431969079922560345576, 5.61187047689635363766717878589, 6.10991189879149770116406898852, 6.21878261684051567398131830055, 6.85115243911160431006956564254, 7.21844494507792221908312935788, 7.24287805417863313466005197587, 7.78019010227179610867979349942, 8.047146849977746708416450894584, 8.230277897450109010067023540050, 8.537336106092434049922447047236, 8.934797914532289439502647803643, 8.961837451665461991740794590060, 9.190149984781258471495861794573, 9.963628164375895277529852286287, 9.964095571279687791110569319213

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