Properties

Label 8-2100e4-1.1-c1e4-0-3
Degree $8$
Conductor $1.945\times 10^{13}$
Sign $1$
Analytic cond. $79065.2$
Root an. cond. $4.09494$
Motivic weight $1$
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  − 2·3-s − 2·7-s + 9-s + 20·13-s + 2·19-s + 4·21-s − 12·23-s + 2·27-s + 8·31-s − 10·37-s − 40·39-s − 24·41-s − 16·43-s − 12·47-s + 7·49-s − 4·57-s − 12·59-s + 2·61-s − 2·63-s + 2·67-s + 24·69-s − 10·73-s − 10·79-s − 4·81-s + 24·83-s − 24·89-s − 40·91-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s + 1/3·9-s + 5.54·13-s + 0.458·19-s + 0.872·21-s − 2.50·23-s + 0.384·27-s + 1.43·31-s − 1.64·37-s − 6.40·39-s − 3.74·41-s − 2.43·43-s − 1.75·47-s + 49-s − 0.529·57-s − 1.56·59-s + 0.256·61-s − 0.251·63-s + 0.244·67-s + 2.88·69-s − 1.17·73-s − 1.12·79-s − 4/9·81-s + 2.63·83-s − 2.54·89-s − 4.19·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(79065.2\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2175960049\)
\(L(\frac12)\) \(\approx\) \(0.2175960049\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( ( 1 + T + T^{2} )^{2} \)
5 \( 1 \)
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
good11$C_2^3$ \( 1 - 4 T^{2} - 105 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
17$C_2^3$ \( 1 - 16 T^{2} - 33 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 - 2 T - 17 T^{2} + 34 T^{3} + 4 T^{4} + 34 p T^{5} - 17 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
37$C_2^2$ \( ( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 12 T + 32 T^{2} + 216 T^{3} + 3567 T^{4} + 216 p T^{5} + 32 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 88 T^{2} + 4935 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 12 T + 8 T^{2} + 216 T^{3} + 6519 T^{4} + 216 p T^{5} + 8 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
67$D_4\times C_2$ \( 1 - 2 T + 31 T^{2} + 322 T^{3} - 4028 T^{4} + 322 p T^{5} + 31 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 10 T + T^{2} - 470 T^{3} - 2828 T^{4} - 470 p T^{5} + p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 10 T - 65 T^{2} + 70 T^{3} + 13084 T^{4} + 70 p T^{5} - 65 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 - 12 T + 184 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 24 T + 272 T^{2} + 3024 T^{3} + 33231 T^{4} + 3024 p T^{5} + 272 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.40572158198811388845679526116, −6.35640873157132417783899449583, −6.09067505299997001892369289197, −5.88381163859989463383228100256, −5.68232301340117280218450214396, −5.57424570824089755967510246634, −5.26347645184539451982804777445, −5.13952074326151028061323926432, −4.82168779665122227955916211929, −4.58065053183296507137583826285, −4.10791413320835240916775449378, −4.09856259763820800582322716322, −3.86456344269450293842362173744, −3.63445347801571816036582299094, −3.34234941806207302239285696847, −3.30447409891624731062376105300, −2.98474637795031599735636426130, −2.97875055148252543778735527489, −2.14100546230502587804246313373, −1.80768544476821815237300552685, −1.64504227890542354468978970161, −1.35803042678977932184805193947, −1.32134429660738897706380508097, −0.71100692373952625411500613478, −0.10060649263523545715549257299, 0.10060649263523545715549257299, 0.71100692373952625411500613478, 1.32134429660738897706380508097, 1.35803042678977932184805193947, 1.64504227890542354468978970161, 1.80768544476821815237300552685, 2.14100546230502587804246313373, 2.97875055148252543778735527489, 2.98474637795031599735636426130, 3.30447409891624731062376105300, 3.34234941806207302239285696847, 3.63445347801571816036582299094, 3.86456344269450293842362173744, 4.09856259763820800582322716322, 4.10791413320835240916775449378, 4.58065053183296507137583826285, 4.82168779665122227955916211929, 5.13952074326151028061323926432, 5.26347645184539451982804777445, 5.57424570824089755967510246634, 5.68232301340117280218450214396, 5.88381163859989463383228100256, 6.09067505299997001892369289197, 6.35640873157132417783899449583, 6.40572158198811388845679526116

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