Properties

Label 8-825e4-1.1-c3e4-0-5
Degree $8$
Conductor $463250390625$
Sign $1$
Analytic cond. $5.61409\times 10^{6}$
Root an. cond. $6.97686$
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  + 23·4-s − 18·9-s − 44·11-s + 273·16-s + 340·19-s + 316·29-s + 120·31-s − 414·36-s + 76·41-s − 1.01e3·44-s + 1.22e3·49-s − 496·59-s + 144·61-s + 1.86e3·64-s + 4.12e3·71-s + 7.82e3·76-s − 1.28e3·79-s + 243·81-s − 488·89-s + 792·99-s + 1.18e3·101-s + 2.01e3·109-s + 7.26e3·116-s + 1.21e3·121-s + 2.76e3·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 23/8·4-s − 2/3·9-s − 1.20·11-s + 4.26·16-s + 4.10·19-s + 2.02·29-s + 0.695·31-s − 1.91·36-s + 0.289·41-s − 3.46·44-s + 3.58·49-s − 1.09·59-s + 0.302·61-s + 3.63·64-s + 6.88·71-s + 11.8·76-s − 1.82·79-s + 1/3·81-s − 0.581·89-s + 0.804·99-s + 1.16·101-s + 1.77·109-s + 5.81·116-s + 0.909·121-s + 1.99·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(5.61409\times 10^{6}\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(26.48167016\)
\(L(\frac12)\) \(\approx\) \(26.48167016\)
\(L(\frac{5}{2})\) 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)$
bad3$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
5 \( 1 \)
11$C_1$ \( ( 1 + p T )^{4} \)
good2$D_4\times C_2$ \( 1 - 23 T^{2} + p^{8} T^{4} - 23 p^{6} T^{6} + p^{12} T^{8} \)
7$D_4\times C_2$ \( 1 - 1228 T^{2} + 611206 T^{4} - 1228 p^{6} T^{6} + p^{12} T^{8} \)
13$D_4\times C_2$ \( 1 - 4704 T^{2} + 15047822 T^{4} - 4704 p^{6} T^{6} + p^{12} T^{8} \)
17$D_4\times C_2$ \( 1 - 3068 T^{2} + 48521926 T^{4} - 3068 p^{6} T^{6} + p^{12} T^{8} \)
19$D_{4}$ \( ( 1 - 170 T + 994 p T^{2} - 170 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 10380 T^{2} + 87755078 T^{4} - 10380 p^{6} T^{6} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 - 158 T + 50106 T^{2} - 158 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 60 T + 59870 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 26796 T^{2} - 2066568586 T^{4} - 26796 p^{6} T^{6} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 - 38 T + 37410 T^{2} - 38 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 54204 T^{2} - 4022052586 T^{4} - 54204 p^{6} T^{6} + p^{12} T^{8} \)
47$D_4\times C_2$ \( 1 + 6372 T^{2} + 5259974 p^{2} T^{4} + 6372 p^{6} T^{6} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 - 157580 T^{2} + 14163088246 T^{4} - 157580 p^{6} T^{6} + p^{12} T^{8} \)
59$D_{4}$ \( ( 1 + 248 T + 181334 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 72 T - 107850 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 924108 T^{2} + 393808416086 T^{4} - 924108 p^{6} T^{6} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 - 2060 T + 1768494 T^{2} - 2060 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 1409376 T^{2} + 795876735710 T^{4} - 1409376 p^{6} T^{6} + p^{12} T^{8} \)
79$D_{4}$ \( ( 1 + 642 T + 691166 T^{2} + 642 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 703704 T^{2} + 714593530334 T^{4} - 703704 p^{6} T^{6} + p^{12} T^{8} \)
89$D_{4}$ \( ( 1 + 244 T + 1355190 T^{2} + 244 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 2079516 T^{2} + 2725067421254 T^{4} - 2079516 p^{6} T^{6} + p^{12} T^{8} \)
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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

−7.06657404974051795656999885010, −6.61170701674766123987470769893, −6.54128650735784058590856815824, −6.31839091473566131092179525575, −6.16963853289000229626982528889, −5.60775682028971814147111514000, −5.53424029963534709263242785939, −5.42287925055142412297937772806, −5.29828134623063684670295875590, −5.00997838876306786949897323657, −4.55848591655620503068830589161, −4.29112399073061607463657642798, −3.98251697883205898659105989023, −3.43155501171034160453977048251, −3.23792624014631241278749856679, −3.06225517693603393875307386883, −3.04546132905551729018468224439, −2.49037797879548003224072618429, −2.45178039961771338546466069177, −2.08914800236058875932364256952, −1.95780278737256209974146234013, −1.17681491253224488590353459131, −1.09779961072852877610754903335, −0.72278701715807835849541370458, −0.58196040904336874519970965885, 0.58196040904336874519970965885, 0.72278701715807835849541370458, 1.09779961072852877610754903335, 1.17681491253224488590353459131, 1.95780278737256209974146234013, 2.08914800236058875932364256952, 2.45178039961771338546466069177, 2.49037797879548003224072618429, 3.04546132905551729018468224439, 3.06225517693603393875307386883, 3.23792624014631241278749856679, 3.43155501171034160453977048251, 3.98251697883205898659105989023, 4.29112399073061607463657642798, 4.55848591655620503068830589161, 5.00997838876306786949897323657, 5.29828134623063684670295875590, 5.42287925055142412297937772806, 5.53424029963534709263242785939, 5.60775682028971814147111514000, 6.16963853289000229626982528889, 6.31839091473566131092179525575, 6.54128650735784058590856815824, 6.61170701674766123987470769893, 7.06657404974051795656999885010

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