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Properties

Label 8-45e4-1.1-c4e4-0-0
Degree $8$
Conductor $4100625$
Sign $1$
Analytic cond. $468.195$
Root an. cond. $2.15676$
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  − 20·7-s − 440·13-s − 217·16-s + 440·25-s − 4.76e3·31-s + 7.24e3·37-s − 5.24e3·43-s + 200·49-s − 5.51e3·61-s + 7.48e3·67-s − 1.56e4·73-s + 8.80e3·91-s − 2.88e4·97-s + 1.23e4·103-s + 4.34e3·112-s + 1.93e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9.68e4·169-s + 173-s + ⋯
L(s)  = 1  − 0.408·7-s − 2.60·13-s − 0.847·16-s + 0.703·25-s − 4.96·31-s + 5.28·37-s − 2.83·43-s + 0.0832·49-s − 1.48·61-s + 1.66·67-s − 2.93·73-s + 1.06·91-s − 3.06·97-s + 1.16·103-s + 0.345·112-s + 0.132·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 3.38·169-s + 3.34e−5·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(4100625\)    =    \(3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(468.195\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 4100625,\ (\ :2, 2, 2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.06544634050\)
\(L(\frac12)\) \(\approx\) \(0.06544634050\)
\(L(3)\) 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 \( 1 \)
5$C_2^2$ \( 1 - 88 p T^{2} + p^{8} T^{4} \)
good2$C_2^3$ \( 1 + 217 T^{4} + p^{16} T^{8} \)
7$C_2^2$ \( ( 1 + 10 T + 50 T^{2} + 10 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 8 p^{2} T^{2} + p^{8} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 220 T + 24200 T^{2} + 220 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
17$C_2^3$ \( 1 + 13944833602 T^{4} + p^{16} T^{8} \)
19$C_2^2$ \( ( 1 - 221438 T^{2} + p^{8} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 156488845118 T^{4} + p^{16} T^{8} \)
29$C_2^2$ \( ( 1 - 422312 T^{2} + p^{8} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 1192 T + p^{4} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 3620 T + 6552200 T^{2} - 3620 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 3250522 T^{2} + p^{8} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 2620 T + 3432200 T^{2} + 2620 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 47566111009598 T^{4} + p^{16} T^{8} \)
53$C_2^3$ \( 1 - 13468208583998 T^{4} + p^{16} T^{8} \)
59$C_2^2$ \( ( 1 - 23242472 T^{2} + p^{8} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 1378 T + p^{4} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 3740 T + 6993800 T^{2} - 3740 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 3299362 T^{2} + p^{8} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 7810 T + 30498050 T^{2} + 7810 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 77525618 T^{2} + p^{8} T^{4} )^{2} \)
83$C_2^3$ \( 1 - 2054763907459838 T^{4} + p^{16} T^{8} \)
89$C_2^2$ \( ( 1 + 4115518 T^{2} + p^{8} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 14410 T + 103824050 T^{2} + 14410 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
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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

−11.03507760854633909344738761781, −10.79483525086061040599500613850, −10.47840878618667146725695396326, −9.892653905701062447475481478538, −9.773782313688874159720520021203, −9.364497714123708185757166080206, −9.144430301314025351405215853671, −9.132610635312828922147333953141, −8.212172781088721986542772818365, −8.075012410630760603122595591713, −7.50097736048996578703036362525, −7.36333016285127982375697565721, −6.96106612645856761042494444078, −6.75487317430919156946041302187, −5.98293104291306731332285174047, −5.81782662249336006436176634683, −5.18739221065715951231411788372, −4.90164581373244738472725529381, −4.42763178341241295579878515372, −3.98522224085245742366018525960, −3.28788318010167756306454506345, −2.64038935079200603895812353510, −2.32610087388951227424866865486, −1.49249365815959785395890319626, −0.084479142783365268261960287368, 0.084479142783365268261960287368, 1.49249365815959785395890319626, 2.32610087388951227424866865486, 2.64038935079200603895812353510, 3.28788318010167756306454506345, 3.98522224085245742366018525960, 4.42763178341241295579878515372, 4.90164581373244738472725529381, 5.18739221065715951231411788372, 5.81782662249336006436176634683, 5.98293104291306731332285174047, 6.75487317430919156946041302187, 6.96106612645856761042494444078, 7.36333016285127982375697565721, 7.50097736048996578703036362525, 8.075012410630760603122595591713, 8.212172781088721986542772818365, 9.132610635312828922147333953141, 9.144430301314025351405215853671, 9.364497714123708185757166080206, 9.773782313688874159720520021203, 9.892653905701062447475481478538, 10.47840878618667146725695396326, 10.79483525086061040599500613850, 11.03507760854633909344738761781

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