Properties

Label 8-825e4-1.1-c1e4-0-4
Degree $8$
Conductor $463250390625$
Sign $1$
Analytic cond. $1883.32$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 4·3-s + 8·4-s + 16·6-s − 8·7-s + 12·8-s + 8·9-s + 32·12-s − 8·13-s − 32·14-s + 15·16-s − 8·17-s + 32·18-s − 32·21-s + 8·23-s + 48·24-s − 32·26-s + 12·27-s − 64·28-s + 24·29-s + 16·31-s + 16·32-s − 32·34-s + 64·36-s − 32·39-s − 128·42-s − 24·43-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 4·4-s + 6.53·6-s − 3.02·7-s + 4.24·8-s + 8/3·9-s + 9.23·12-s − 2.21·13-s − 8.55·14-s + 15/4·16-s − 1.94·17-s + 7.54·18-s − 6.98·21-s + 1.66·23-s + 9.79·24-s − 6.27·26-s + 2.30·27-s − 12.0·28-s + 4.45·29-s + 2.87·31-s + 2.82·32-s − 5.48·34-s + 32/3·36-s − 5.12·39-s − 19.7·42-s − 3.65·43-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1883.32\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(18.77868669\)
\(L(\frac12)\) \(\approx\) \(18.77868669\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
5 \( 1 \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
good2$D_4\times C_2$ \( 1 - p^{2} T + p^{3} T^{2} - 3 p^{2} T^{3} + 17 T^{4} - 3 p^{3} T^{5} + p^{5} T^{6} - p^{5} T^{7} + p^{4} T^{8} \)
7$C_4\times C_2$ \( 1 + 8 T + 32 T^{2} + 88 T^{3} + 226 T^{4} + 88 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 + 8 T + 32 T^{2} + 136 T^{3} + 562 T^{4} + 136 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 52 T^{2} + 1270 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 8 T + 32 T^{2} - 120 T^{3} + 386 T^{4} - 120 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
37$C_2^3$ \( 1 - 2638 T^{4} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 140 T^{2} + 8134 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 24 T + 288 T^{2} + 2664 T^{3} + 20018 T^{4} + 2664 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 + 8 T + 32 T^{2} + 312 T^{3} + 2978 T^{4} + 312 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 4174 T^{4} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
67$D_4\times C_2$ \( 1 - 16 T + 128 T^{2} - 1520 T^{3} + 17266 T^{4} - 1520 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 188 T^{2} + 16870 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 + 24 T + 288 T^{2} + 3384 T^{3} + 35138 T^{4} + 3384 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 292 T^{2} + 33670 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2^3$ \( 1 + 3122 T^{4} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 16 T + 128 T^{2} + 1040 T^{3} + 7426 T^{4} + 1040 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.10527903962260321306404133691, −7.02345850383876131462580843015, −6.79903078188632046789956573932, −6.52753769437103760566372414465, −6.41518835879857036875369705868, −6.33829448713406888952126886621, −6.24430954658328204813905575526, −5.47440605728043442488593108616, −5.32051823551166086005675653144, −4.89930764233685310221701021580, −4.82690081537264511203168103788, −4.75923039704264644849448770552, −4.28444265971986336945688110293, −4.28032781480575632462535391856, −4.15104863010959845932273260243, −3.31112885931583810721935537207, −3.28781530722005834460899896745, −3.21311261623855890029700422658, −3.03890962679409388841417329251, −2.72545962497927793317136682411, −2.51512906925252361408136591403, −2.33884090271654022187514507164, −1.99685863304220314870868752202, −1.16832028659793016481114662766, −0.51881742896170273844016789002, 0.51881742896170273844016789002, 1.16832028659793016481114662766, 1.99685863304220314870868752202, 2.33884090271654022187514507164, 2.51512906925252361408136591403, 2.72545962497927793317136682411, 3.03890962679409388841417329251, 3.21311261623855890029700422658, 3.28781530722005834460899896745, 3.31112885931583810721935537207, 4.15104863010959845932273260243, 4.28032781480575632462535391856, 4.28444265971986336945688110293, 4.75923039704264644849448770552, 4.82690081537264511203168103788, 4.89930764233685310221701021580, 5.32051823551166086005675653144, 5.47440605728043442488593108616, 6.24430954658328204813905575526, 6.33829448713406888952126886621, 6.41518835879857036875369705868, 6.52753769437103760566372414465, 6.79903078188632046789956573932, 7.02345850383876131462580843015, 7.10527903962260321306404133691

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