Properties

Label 8-200e4-1.1-c5e4-0-1
Degree $8$
Conductor $1600000000$
Sign $1$
Analytic cond. $1.05867\times 10^{6}$
Root an. cond. $5.66363$
Motivic weight $5$
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  + 458·9-s − 400·11-s + 1.68e3·19-s + 9.36e3·29-s − 1.00e4·31-s − 1.06e4·41-s + 6.52e4·49-s + 1.63e5·59-s − 9.38e4·61-s + 1.48e4·71-s + 2.16e5·79-s + 5.46e4·81-s − 1.41e5·89-s − 1.83e5·99-s − 3.02e5·101-s − 2.82e5·109-s − 3.31e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6.61e3·169-s + ⋯
L(s)  = 1  + 1.88·9-s − 0.996·11-s + 1.06·19-s + 2.06·29-s − 1.87·31-s − 0.991·41-s + 3.88·49-s + 6.11·59-s − 3.22·61-s + 0.350·71-s + 3.89·79-s + 0.925·81-s − 1.89·89-s − 1.87·99-s − 2.94·101-s − 2.28·109-s − 2.05·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 0.0178·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.05867\times 10^{6}\)
Root analytic conductor: \(5.66363\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(3.705908790\)
\(L(\frac12)\) \(\approx\) \(3.705908790\)
\(L(\frac{7}{2})\) 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 \( 1 \)
good3$D_4\times C_2$ \( 1 - 458 T^{2} + 1915 p^{4} T^{4} - 458 p^{10} T^{6} + p^{20} T^{8} \)
7$D_4\times C_2$ \( 1 - 1332 p^{2} T^{2} + 1629866758 T^{4} - 1332 p^{12} T^{6} + p^{20} T^{8} \)
11$D_{4}$ \( ( 1 + 200 T + 225821 T^{2} + 200 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 6612 T^{2} + 47343141238 T^{4} - 6612 p^{10} T^{6} + p^{20} T^{8} \)
17$D_4\times C_2$ \( 1 - 5517394 T^{2} + 11637628825043 T^{4} - 5517394 p^{10} T^{6} + p^{20} T^{8} \)
19$D_{4}$ \( ( 1 - 840 T + 4289677 T^{2} - 840 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 19211092 T^{2} + 166300343917958 T^{4} - 19211092 p^{10} T^{6} + p^{20} T^{8} \)
29$D_{4}$ \( ( 1 - 4680 T + 1421346 p T^{2} - 4680 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 5008 T + 33327162 T^{2} + 5008 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 86339436 T^{2} + 2659590797035222 T^{4} - 86339436 p^{10} T^{6} + p^{20} T^{8} \)
41$D_{4}$ \( ( 1 + 5334 T + 27686155 T^{2} + 5334 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 29960652 T^{2} + 43433374484607958 T^{4} - 29960652 p^{10} T^{6} + p^{20} T^{8} \)
47$C_2^2$ \( ( 1 - 288752718 T^{2} + p^{10} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 514633100 T^{2} + 347542581493770198 T^{4} - 514633100 p^{10} T^{6} + p^{20} T^{8} \)
59$D_{4}$ \( ( 1 - 81776 T + 3059970646 T^{2} - 81776 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 46932 T + 2239379182 T^{2} + 46932 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 2982943418 T^{2} + 5751121538625041883 T^{4} - 2982943418 p^{10} T^{6} + p^{20} T^{8} \)
71$D_{4}$ \( ( 1 - 7448 T + 3593807902 T^{2} - 7448 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 2368673842 T^{2} + 9983110585747236243 T^{4} - 2368673842 p^{10} T^{6} + p^{20} T^{8} \)
79$D_{4}$ \( ( 1 - 108104 T + 6816153098 T^{2} - 108104 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 8274508042 T^{2} + 45513777799710942443 T^{4} - 8274508042 p^{10} T^{6} + p^{20} T^{8} \)
89$D_{4}$ \( ( 1 + 70990 T + 6345035107 T^{2} + 70990 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 22699126076 T^{2} + \)\(24\!\cdots\!42\)\( T^{4} - 22699126076 p^{10} T^{6} + p^{20} T^{8} \)
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

−8.286811720222808775627123115833, −7.78168189526052989049280944731, −7.51676253490780307409864631069, −7.43987228965259597004839371500, −7.12470316782803923429317387202, −6.88563125940497570018097100161, −6.53995649664616726751603688828, −6.52789886865690339368444716023, −5.76660573415419976128301052032, −5.64131624672435032356338368571, −5.27122104770866845200703011982, −5.03190316312084037726302603822, −4.99296988038484382731376090583, −4.19879431243808808820764684618, −4.16280880924102402205355681774, −3.75403274901326919601509234522, −3.68841743811035441554087435286, −2.98920402081083469267602018400, −2.57611695377271066608189571721, −2.41179965199944009204787782543, −2.01029974440302141693590370904, −1.34224616212102895751581717007, −1.15156384664476019735896828331, −0.826974206065066676624359460516, −0.27059798468726044556933186956, 0.27059798468726044556933186956, 0.826974206065066676624359460516, 1.15156384664476019735896828331, 1.34224616212102895751581717007, 2.01029974440302141693590370904, 2.41179965199944009204787782543, 2.57611695377271066608189571721, 2.98920402081083469267602018400, 3.68841743811035441554087435286, 3.75403274901326919601509234522, 4.16280880924102402205355681774, 4.19879431243808808820764684618, 4.99296988038484382731376090583, 5.03190316312084037726302603822, 5.27122104770866845200703011982, 5.64131624672435032356338368571, 5.76660573415419976128301052032, 6.52789886865690339368444716023, 6.53995649664616726751603688828, 6.88563125940497570018097100161, 7.12470316782803923429317387202, 7.43987228965259597004839371500, 7.51676253490780307409864631069, 7.78168189526052989049280944731, 8.286811720222808775627123115833

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