Properties

Label 8-720e4-1.1-c3e4-0-8
Degree $8$
Conductor $268738560000$
Sign $1$
Analytic cond. $3.25682\times 10^{6}$
Root an. cond. $6.51777$
Motivic weight $3$
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  − 6·5-s − 84·11-s + 112·19-s + 146·25-s − 636·29-s − 104·31-s + 816·41-s + 616·49-s + 504·55-s + 372·59-s + 680·61-s − 72·71-s + 760·79-s + 2.23e3·89-s − 672·95-s + 2.24e3·101-s + 1.32e3·109-s − 176·121-s − 2.28e3·125-s + 127-s + 131-s + 137-s + 139-s + 3.81e3·145-s + 149-s + 151-s + 624·155-s + ⋯
L(s)  = 1  − 0.536·5-s − 2.30·11-s + 1.35·19-s + 1.16·25-s − 4.07·29-s − 0.602·31-s + 3.10·41-s + 1.79·49-s + 1.23·55-s + 0.820·59-s + 1.42·61-s − 0.120·71-s + 1.08·79-s + 2.65·89-s − 0.725·95-s + 2.21·101-s + 1.16·109-s − 0.132·121-s − 1.63·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 2.18·145-s + 0.000549·149-s + 0.000538·151-s + 0.323·155-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(3.25682\times 10^{6}\)
Root analytic conductor: \(6.51777\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} \cdot 5^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(4.538274207\)
\(L(\frac12)\) \(\approx\) \(4.538274207\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_2^2$ \( 1 + 6 T - 22 p T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
good7$D_4\times C_2$ \( 1 - 88 p T^{2} + 316878 T^{4} - 88 p^{7} T^{6} + p^{12} T^{8} \)
11$D_{4}$ \( ( 1 + 42 T + 2734 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 5008 T^{2} + 13678638 T^{4} - 5008 p^{6} T^{6} + p^{12} T^{8} \)
17$D_4\times C_2$ \( 1 - 12400 T^{2} + 76051038 T^{4} - 12400 p^{6} T^{6} + p^{12} T^{8} \)
19$D_{4}$ \( ( 1 - 56 T + 8598 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 46204 T^{2} + 828262758 T^{4} - 46204 p^{6} T^{6} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 + 318 T + 55978 T^{2} + 318 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 52 T + 58782 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 96016 T^{2} + 4637534478 T^{4} - 96016 p^{6} T^{6} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 - 408 T + 177982 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 121900 T^{2} + 14997346998 T^{4} - 121900 p^{6} T^{6} + p^{12} T^{8} \)
47$D_4\times C_2$ \( 1 - 225580 T^{2} + 31469120358 T^{4} - 225580 p^{6} T^{6} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 - 376864 T^{2} + 68697431598 T^{4} - 376864 p^{6} T^{6} + p^{12} T^{8} \)
59$D_{4}$ \( ( 1 - 186 T + 419038 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 340 T + 388398 T^{2} - 340 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 861340 T^{2} + 346621507638 T^{4} - 861340 p^{6} T^{6} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 + 36 T + 384046 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 429844 T^{2} + 136740794118 T^{4} - 429844 p^{6} T^{6} + p^{12} T^{8} \)
79$D_{4}$ \( ( 1 - 380 T + 99678 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 916108 T^{2} + 434315280918 T^{4} - 916108 p^{6} T^{6} + p^{12} T^{8} \)
89$D_{4}$ \( ( 1 - 1116 T + 1508758 T^{2} - 1116 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 2174980 T^{2} + 2500346420358 T^{4} - 2174980 p^{6} T^{6} + p^{12} T^{8} \)
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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

−7.32760523999665518595100334166, −6.74175498064494535197159883119, −6.69424388271704429931121888263, −6.50241972965961002212713433313, −5.89330648341779241396589488719, −5.87709887187600333241532550495, −5.49485973972768383863281247451, −5.32839299599788793662937447165, −5.28200316590202535669379322091, −5.16854727168092994149537940102, −4.65462759898636209791835492958, −4.28093030713821069968264573978, −4.15306207506281699291302613090, −3.71213817882796943753970028791, −3.71033914881123060810941537211, −3.28430106088384710439753590585, −3.04933660917032998174598126845, −2.54704828842364215306893214284, −2.41318541054096188250555171794, −2.29593398534495789170469447916, −1.69981276230065148776252260041, −1.50028072794779841463571661946, −0.64100420281826216527093165951, −0.55258587749737921052581322302, −0.50083597825191823173676834139, 0.50083597825191823173676834139, 0.55258587749737921052581322302, 0.64100420281826216527093165951, 1.50028072794779841463571661946, 1.69981276230065148776252260041, 2.29593398534495789170469447916, 2.41318541054096188250555171794, 2.54704828842364215306893214284, 3.04933660917032998174598126845, 3.28430106088384710439753590585, 3.71033914881123060810941537211, 3.71213817882796943753970028791, 4.15306207506281699291302613090, 4.28093030713821069968264573978, 4.65462759898636209791835492958, 5.16854727168092994149537940102, 5.28200316590202535669379322091, 5.32839299599788793662937447165, 5.49485973972768383863281247451, 5.87709887187600333241532550495, 5.89330648341779241396589488719, 6.50241972965961002212713433313, 6.69424388271704429931121888263, 6.74175498064494535197159883119, 7.32760523999665518595100334166

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