Properties

Label 8-1050e4-1.1-c1e4-0-27
Degree $8$
Conductor $1.216\times 10^{12}$
Sign $1$
Analytic cond. $4941.57$
Root an. cond. $2.89556$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 6·3-s + 4-s + 12·6-s + 2·8-s + 21·9-s − 6·12-s − 4·16-s − 12·17-s − 42·18-s − 12·19-s − 12·23-s − 12·24-s − 54·27-s − 6·31-s + 2·32-s + 24·34-s + 21·36-s + 24·38-s + 24·46-s − 24·47-s + 24·48-s − 11·49-s + 72·51-s − 18·53-s + 108·54-s + 72·57-s + ⋯
L(s)  = 1  − 1.41·2-s − 3.46·3-s + 1/2·4-s + 4.89·6-s + 0.707·8-s + 7·9-s − 1.73·12-s − 16-s − 2.91·17-s − 9.89·18-s − 2.75·19-s − 2.50·23-s − 2.44·24-s − 10.3·27-s − 1.07·31-s + 0.353·32-s + 4.11·34-s + 7/2·36-s + 3.89·38-s + 3.53·46-s − 3.50·47-s + 3.46·48-s − 1.57·49-s + 10.0·51-s − 2.47·53-s + 14.6·54-s + 9.53·57-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(4941.57\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T + T^{2} )^{2} \)
3$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
5 \( 1 \)
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
good11$C_2^3$ \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( ( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
37$C_2^2$$\times$$C_2^2$ \( ( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \)
41$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 12 T + 95 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 - 115 T^{2} + 9744 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^3$ \( 1 - 98 T^{2} + 4275 T^{4} - 98 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 91 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 + 167 T^{2} + p^{2} T^{4} )^{2} \)
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

−7.73797270931369872945207767938, −7.00875333003205074510190238527, −6.85760198257489910986611233812, −6.85119087835129809259418569431, −6.68975243829269091273780813837, −6.39551878191952648370641495400, −6.17405697917139266517252019248, −6.07794352423813635703091143968, −6.05699670719865008516638579394, −5.61605459286275017501719482348, −5.24494937344771526425172908541, −4.96170068342187750989397367121, −4.79196407689664757775744195782, −4.71383007952415188173498076501, −4.50345347372588764675566966873, −4.24818510994907391555937554864, −3.93863113598051416985130476312, −3.63324165221143756356840086905, −3.62745866484663030892298841633, −2.60977514441919106655771403569, −2.49790404754263536251347492562, −1.98251300528504065055767943605, −1.76511693994370168460518702156, −1.51577908865055299190380275351, −1.23624692067507408870995222339, 0, 0, 0, 0, 1.23624692067507408870995222339, 1.51577908865055299190380275351, 1.76511693994370168460518702156, 1.98251300528504065055767943605, 2.49790404754263536251347492562, 2.60977514441919106655771403569, 3.62745866484663030892298841633, 3.63324165221143756356840086905, 3.93863113598051416985130476312, 4.24818510994907391555937554864, 4.50345347372588764675566966873, 4.71383007952415188173498076501, 4.79196407689664757775744195782, 4.96170068342187750989397367121, 5.24494937344771526425172908541, 5.61605459286275017501719482348, 6.05699670719865008516638579394, 6.07794352423813635703091143968, 6.17405697917139266517252019248, 6.39551878191952648370641495400, 6.68975243829269091273780813837, 6.85119087835129809259418569431, 6.85760198257489910986611233812, 7.00875333003205074510190238527, 7.73797270931369872945207767938

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