Properties

Label 8-189e4-1.1-c1e4-0-4
Degree $8$
Conductor $1275989841$
Sign $1$
Analytic cond. $5.18747$
Root an. cond. $1.22848$
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  + 4·4-s + 4·7-s + 4·16-s − 14·25-s + 16·28-s + 20·37-s + 20·43-s − 2·49-s − 16·64-s + 8·67-s − 52·79-s − 56·100-s − 28·109-s + 16·112-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 80·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 80·172-s + ⋯
L(s)  = 1  + 2·4-s + 1.51·7-s + 16-s − 2.79·25-s + 3.02·28-s + 3.28·37-s + 3.04·43-s − 2/7·49-s − 2·64-s + 0.977·67-s − 5.85·79-s − 5.59·100-s − 2.68·109-s + 1.51·112-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 6.57·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.07·169-s + 6.09·172-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{12} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(5.18747\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{189} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.863349295\)
\(L(\frac12)\) \(\approx\) \(2.863349295\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
good2$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - p T^{2} )^{4} \)
79$C_2$ \( ( 1 + 13 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 163 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 100 T^{2} + p^{2} T^{4} )^{2} \)
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

−9.341091374372791732670732620300, −8.909766763618312675979689306802, −8.507092058131594793711533661069, −8.362479658156968219551841550469, −8.006987996081415846114091344901, −7.72969249655452372306215068052, −7.57985669344396046469323545879, −7.42781759225687197034043682863, −7.07752398678913708314848478297, −6.75682140315435819717536517767, −6.49192134645838242014543461484, −6.02741144301485468951493417722, −5.81826367315714878428283556666, −5.62524808166462633392988730764, −5.60233471199409282489903453001, −4.63727216477519667024161427335, −4.47567587387250422591413427531, −4.27294115020486697734705743889, −4.06322499893154788370364151799, −3.26072537580287945895711806105, −2.87821755845660599453994694347, −2.50285706865570168890423261532, −2.11618973299712712027780402745, −1.83112991167314617853490303370, −1.21390625503354902557172281617, 1.21390625503354902557172281617, 1.83112991167314617853490303370, 2.11618973299712712027780402745, 2.50285706865570168890423261532, 2.87821755845660599453994694347, 3.26072537580287945895711806105, 4.06322499893154788370364151799, 4.27294115020486697734705743889, 4.47567587387250422591413427531, 4.63727216477519667024161427335, 5.60233471199409282489903453001, 5.62524808166462633392988730764, 5.81826367315714878428283556666, 6.02741144301485468951493417722, 6.49192134645838242014543461484, 6.75682140315435819717536517767, 7.07752398678913708314848478297, 7.42781759225687197034043682863, 7.57985669344396046469323545879, 7.72969249655452372306215068052, 8.006987996081415846114091344901, 8.362479658156968219551841550469, 8.507092058131594793711533661069, 8.909766763618312675979689306802, 9.341091374372791732670732620300

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