Properties

Label 8-896e4-1.1-c1e4-0-10
Degree $8$
Conductor $644513529856$
Sign $1$
Analytic cond. $2620.23$
Root an. cond. $2.67480$
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  − 4·5-s + 8·7-s + 4·9-s − 8·11-s − 12·13-s − 4·25-s − 32·35-s + 24·43-s − 16·45-s + 34·49-s + 32·55-s + 12·61-s + 32·63-s + 48·65-s − 24·67-s − 64·77-s + 6·81-s − 96·91-s − 32·99-s + 4·101-s + 48·103-s + 8·107-s − 24·113-s − 48·117-s + 20·121-s + 44·125-s + 127-s + ⋯
L(s)  = 1  − 1.78·5-s + 3.02·7-s + 4/3·9-s − 2.41·11-s − 3.32·13-s − 4/5·25-s − 5.40·35-s + 3.65·43-s − 2.38·45-s + 34/7·49-s + 4.31·55-s + 1.53·61-s + 4.03·63-s + 5.95·65-s − 2.93·67-s − 7.29·77-s + 2/3·81-s − 10.0·91-s − 3.21·99-s + 0.398·101-s + 4.72·103-s + 0.773·107-s − 2.25·113-s − 4.43·117-s + 1.81·121-s + 3.93·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2620.23\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{896} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.737250394\)
\(L(\frac12)\) \(\approx\) \(1.737250394\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
good3$D_4\times C_2$ \( 1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
5$D_{4}$ \( ( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$D_{4}$ \( ( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_{4}$ \( ( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 52 T^{2} + 1290 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 36 T^{2} + 614 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 20 T^{2} - 1686 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 - 6 T + 56 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 132 T^{2} + 11366 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 164 T^{2} + 19530 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 132 T^{2} + 7910 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
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

−7.38232468742527731645632474376, −7.23175189388237959629776825590, −7.22012803478872992568963484150, −6.78954321362138876887089126078, −6.40750983352129277576636035264, −5.93077526664128178923172511820, −5.70067820394470215721721253687, −5.42861085727568203242823398289, −5.41316743239096358681565202596, −5.12448716454450108178972077456, −4.72819708528956310774108383687, −4.63742678297712441093110923420, −4.46107299867152544962790831716, −4.23032418459533185595911699425, −4.21856786524656515731551904812, −3.84777846204173715390377319389, −3.18364242530950448584662721867, −3.10497425902761634594239660906, −2.62089834895025186585259445390, −2.30632300220758869695322856539, −2.07947298246006322917760616113, −1.97771810361146053890213385609, −1.50965111345812625728243321209, −0.66277459923089049582352058904, −0.45624222994416594933553733501, 0.45624222994416594933553733501, 0.66277459923089049582352058904, 1.50965111345812625728243321209, 1.97771810361146053890213385609, 2.07947298246006322917760616113, 2.30632300220758869695322856539, 2.62089834895025186585259445390, 3.10497425902761634594239660906, 3.18364242530950448584662721867, 3.84777846204173715390377319389, 4.21856786524656515731551904812, 4.23032418459533185595911699425, 4.46107299867152544962790831716, 4.63742678297712441093110923420, 4.72819708528956310774108383687, 5.12448716454450108178972077456, 5.41316743239096358681565202596, 5.42861085727568203242823398289, 5.70067820394470215721721253687, 5.93077526664128178923172511820, 6.40750983352129277576636035264, 6.78954321362138876887089126078, 7.22012803478872992568963484150, 7.23175189388237959629776825590, 7.38232468742527731645632474376

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