Properties

Label 8-1950e4-1.1-c1e4-0-31
Degree $8$
Conductor $1.446\times 10^{13}$
Sign $1$
Analytic cond. $58782.3$
Root an. cond. $3.94598$
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-s + 9-s − 2·11-s − 10·19-s + 40·31-s + 36-s − 12·41-s − 2·44-s − 5·49-s + 8·59-s + 4·61-s − 64-s + 4·71-s − 10·76-s + 40·79-s + 2·89-s − 2·99-s − 8·101-s + 40·109-s + 23·121-s + 40·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1/2·4-s + 1/3·9-s − 0.603·11-s − 2.29·19-s + 7.18·31-s + 1/6·36-s − 1.87·41-s − 0.301·44-s − 5/7·49-s + 1.04·59-s + 0.512·61-s − 1/8·64-s + 0.474·71-s − 1.14·76-s + 4.50·79-s + 0.211·89-s − 0.201·99-s − 0.796·101-s + 3.83·109-s + 2.09·121-s + 3.59·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(58782.3\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1950} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.888630841\)
\(L(\frac12)\) \(\approx\) \(4.888630841\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3$C_2^2$ \( 1 - T^{2} + T^{4} \)
5 \( 1 \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
good7$C_2^3$ \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
37$C_2^2$$\times$$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \)
41$C_2^2$ \( ( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^3$ \( 1 + 82 T^{2} + 4875 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 63 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 - 10 T^{2} - 4389 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - 2 T - 67 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - T - 88 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^3$ \( 1 + 50 T^{2} - 6909 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \)
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

−6.54173596906625829547403904917, −6.34722435401084913572718038489, −6.24281797698780537196447559289, −6.22787042431572163422041336642, −5.76738666415937700560585516389, −5.27757529334371296657821366620, −5.24288394262853844080481793506, −4.98649936128342067461450213170, −4.85568446522594580590303025835, −4.63211044674775159380957326690, −4.30005960285125334388322367736, −4.25954523239048061148651515877, −4.03341269987190224942214397921, −3.63479778008292596754054398793, −3.38218667500963820800756458191, −3.07827941879822769644181305241, −2.90671106654086169811173202293, −2.59444830673061003757048273862, −2.32851210680820132126305005842, −2.18209457349001252824412030441, −2.02116409797950001818676701295, −1.41013736048468723594731439097, −1.13994476464709636524394066693, −0.69313364994574397675756963150, −0.47984753263988663761659026834, 0.47984753263988663761659026834, 0.69313364994574397675756963150, 1.13994476464709636524394066693, 1.41013736048468723594731439097, 2.02116409797950001818676701295, 2.18209457349001252824412030441, 2.32851210680820132126305005842, 2.59444830673061003757048273862, 2.90671106654086169811173202293, 3.07827941879822769644181305241, 3.38218667500963820800756458191, 3.63479778008292596754054398793, 4.03341269987190224942214397921, 4.25954523239048061148651515877, 4.30005960285125334388322367736, 4.63211044674775159380957326690, 4.85568446522594580590303025835, 4.98649936128342067461450213170, 5.24288394262853844080481793506, 5.27757529334371296657821366620, 5.76738666415937700560585516389, 6.22787042431572163422041336642, 6.24281797698780537196447559289, 6.34722435401084913572718038489, 6.54173596906625829547403904917

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