Properties

Label 8-1536e4-1.1-c1e4-0-16
Degree $8$
Conductor $5.566\times 10^{12}$
Sign $1$
Analytic cond. $22629.4$
Root an. cond. $3.50214$
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 + 12·13-s − 16·17-s + 8·25-s − 4·29-s + 12·37-s + 12·49-s + 20·53-s − 4·61-s − 48·65-s − 81-s + 64·85-s + 64·97-s − 44·101-s − 12·109-s + 72·113-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1.78·5-s + 3.32·13-s − 3.88·17-s + 8/5·25-s − 0.742·29-s + 1.97·37-s + 12/7·49-s + 2.74·53-s − 0.512·61-s − 5.95·65-s − 1/9·81-s + 6.94·85-s + 6.49·97-s − 4.37·101-s − 1.14·109-s + 6.77·113-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{36} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(22629.4\)
Root analytic conductor: \(3.50214\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1536} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{36} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.389816997\)
\(L(\frac12)\) \(\approx\) \(2.389816997\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + T^{4} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
7$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
19$C_2^3$ \( 1 - 46 T^{4} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
59$C_2^3$ \( 1 - 6286 T^{4} + p^{4} T^{8} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
67$C_2^3$ \( 1 + 4946 T^{4} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 - 5678 T^{4} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 16 T + p T^{2} )^{4} \)
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

−6.60735782479540872737213519337, −6.60533721879017309981301527491, −6.33751104689374035904444603679, −6.12893948418251827249722409462, −5.99556195027081780536110672705, −5.74941942957228156403274914040, −5.40793527245946670033377468939, −5.35399373114447173072700966105, −4.65128054919586659214019225341, −4.60142970613711549972779290239, −4.59892000310744603301922542136, −4.12410289797432858631407537633, −4.08532292656413165095716198861, −3.87330137131715963396298802163, −3.63854250979848295248803949735, −3.54196023554807279746337472450, −3.03799731554637758135619553233, −2.82740154341321765283510203429, −2.52744501788909875590657922845, −2.10716484483465595899904881848, −1.92860737206447420557457776992, −1.66951985553575565823282547879, −0.865745610145725716890159854420, −0.819596269594554074778845238395, −0.42234843093892868677218902358, 0.42234843093892868677218902358, 0.819596269594554074778845238395, 0.865745610145725716890159854420, 1.66951985553575565823282547879, 1.92860737206447420557457776992, 2.10716484483465595899904881848, 2.52744501788909875590657922845, 2.82740154341321765283510203429, 3.03799731554637758135619553233, 3.54196023554807279746337472450, 3.63854250979848295248803949735, 3.87330137131715963396298802163, 4.08532292656413165095716198861, 4.12410289797432858631407537633, 4.59892000310744603301922542136, 4.60142970613711549972779290239, 4.65128054919586659214019225341, 5.35399373114447173072700966105, 5.40793527245946670033377468939, 5.74941942957228156403274914040, 5.99556195027081780536110672705, 6.12893948418251827249722409462, 6.33751104689374035904444603679, 6.60533721879017309981301527491, 6.60735782479540872737213519337

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