Properties

Label 8-320e4-1.1-c1e4-0-1
Degree $8$
Conductor $10485760000$
Sign $1$
Analytic cond. $42.6293$
Root an. cond. $1.59850$
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  − 12·7-s + 4·9-s − 12·23-s − 2·25-s + 24·31-s + 24·41-s + 12·47-s + 68·49-s − 48·63-s − 24·71-s − 16·73-s + 48·79-s + 6·81-s − 24·89-s + 16·97-s − 12·103-s − 24·113-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 144·161-s + 163-s + ⋯
L(s)  = 1  − 4.53·7-s + 4/3·9-s − 2.50·23-s − 2/5·25-s + 4.31·31-s + 3.74·41-s + 1.75·47-s + 68/7·49-s − 6.04·63-s − 2.84·71-s − 1.87·73-s + 5.40·79-s + 2/3·81-s − 2.54·89-s + 1.62·97-s − 1.18·103-s − 2.25·113-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 11.3·161-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(42.6293\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{320} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.8849732351\)
\(L(\frac12)\) \(\approx\) \(0.8849732351\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$D_4\times C_2$ \( 1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
7$D_{4}$ \( ( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
23$D_{4}$ \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{4} \)
31$D_{4}$ \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 68 T^{2} + 2154 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - 76 T^{2} + 1974 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 164 T^{2} + 13002 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
83$D_4\times C_2$ \( 1 - 308 T^{2} + 37386 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−8.389806534722554268650255506042, −8.168913389320729669025370928443, −7.892735705835243592516388586619, −7.78190255295187971630639995418, −7.18198777234494538764232766986, −7.09376537108382686286186702029, −7.01517144691756090404615941023, −6.48637670656232606959699721908, −6.38324784577413286092374057680, −6.11781683976858902825569331369, −6.03801148232836937886084108644, −5.81473091863710290136044589843, −5.63671100692969905969956586197, −4.72977610001578692730313778408, −4.56466113048466223402066765080, −4.24128575666684809197801284385, −4.07204432030604953258822973427, −3.70419619239452920187104191926, −3.46438869456891020535988927813, −2.90466096968396632598197225311, −2.79817035857445385143577081172, −2.55409254358087990562653608006, −2.01877446154662077728623416603, −0.997678948698118200807096413095, −0.50813332105165067251128578781, 0.50813332105165067251128578781, 0.997678948698118200807096413095, 2.01877446154662077728623416603, 2.55409254358087990562653608006, 2.79817035857445385143577081172, 2.90466096968396632598197225311, 3.46438869456891020535988927813, 3.70419619239452920187104191926, 4.07204432030604953258822973427, 4.24128575666684809197801284385, 4.56466113048466223402066765080, 4.72977610001578692730313778408, 5.63671100692969905969956586197, 5.81473091863710290136044589843, 6.03801148232836937886084108644, 6.11781683976858902825569331369, 6.38324784577413286092374057680, 6.48637670656232606959699721908, 7.01517144691756090404615941023, 7.09376537108382686286186702029, 7.18198777234494538764232766986, 7.78190255295187971630639995418, 7.892735705835243592516388586619, 8.168913389320729669025370928443, 8.389806534722554268650255506042

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