Properties

Label 8-252e4-1.1-c1e4-0-4
Degree $8$
Conductor $4032758016$
Sign $1$
Analytic cond. $16.3949$
Root an. cond. $1.41853$
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  + 2-s + 4-s − 2·7-s + 3·8-s − 2·14-s + 16-s − 12·19-s + 6·25-s − 2·28-s + 8·29-s − 32-s − 12·37-s − 12·38-s − 8·47-s + 6·49-s + 6·50-s − 16·53-s − 6·56-s + 8·58-s + 16·59-s + 64-s − 12·74-s − 12·76-s − 8·83-s − 8·94-s + 6·98-s + 6·100-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.755·7-s + 1.06·8-s − 0.534·14-s + 1/4·16-s − 2.75·19-s + 6/5·25-s − 0.377·28-s + 1.48·29-s − 0.176·32-s − 1.97·37-s − 1.94·38-s − 1.16·47-s + 6/7·49-s + 0.848·50-s − 2.19·53-s − 0.801·56-s + 1.05·58-s + 2.08·59-s + 1/8·64-s − 1.39·74-s − 1.37·76-s − 0.878·83-s − 0.825·94-s + 0.606·98-s + 3/5·100-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(16.3949\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{252} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.150534770\)
\(L(\frac12)\) \(\approx\) \(2.150534770\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
3 \( 1 \)
7$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
good5$C_2^2 \wr C_2$ \( 1 - 6 T^{2} + 42 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2 \wr C_2$ \( 1 - 34 T^{2} + 514 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2 \wr C_2$ \( 1 - 12 T^{2} + 102 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2 \wr C_2$ \( 1 - 22 T^{2} + 682 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - 82 T^{2} + 2722 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$D_{4}$ \( ( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - 102 T^{2} + 5130 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2^2 \wr C_2$ \( 1 - 24 T^{2} + 3774 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
61$C_2^2 \wr C_2$ \( 1 - 132 T^{2} + 10710 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2 \wr C_2$ \( 1 - 144 T^{2} + 10830 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2 \wr C_2$ \( 1 - 114 T^{2} + 8418 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^2 \wr C_2$ \( 1 - 236 T^{2} + 24310 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2 \wr C_2$ \( 1 - 288 T^{2} + 33150 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
89$C_2^2 \wr C_2$ \( 1 - 294 T^{2} + 36618 T^{4} - 294 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2 \wr C_2$ \( 1 - 204 T^{2} + 28950 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \)
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

−8.716570320669932295560392336736, −8.561676929671817800086832661022, −8.199201998403981279907580734221, −8.112271176030091713646032641211, −7.910688366249232273735482437063, −7.14431026890047472265958664964, −6.96874588769461866320379606334, −6.96528112629200188798302820996, −6.73505616909807984622477097375, −6.57083924130414393763449692854, −5.92388708571680527569796532004, −5.78057953225196759515243928198, −5.77868922247834442174551932415, −5.08843144426301272963333119371, −4.69304626545211694618803372859, −4.54999947629504821640751486100, −4.46711452391305228634967994700, −4.10920113436614441052851173917, −3.39538406667254509395169331491, −3.20508308914145837452820012099, −3.17823182755974289963550278791, −2.30709495294607646192895702078, −2.00998914018690913983320959374, −1.79586169949253477839784317123, −0.66884033543823757728129316142, 0.66884033543823757728129316142, 1.79586169949253477839784317123, 2.00998914018690913983320959374, 2.30709495294607646192895702078, 3.17823182755974289963550278791, 3.20508308914145837452820012099, 3.39538406667254509395169331491, 4.10920113436614441052851173917, 4.46711452391305228634967994700, 4.54999947629504821640751486100, 4.69304626545211694618803372859, 5.08843144426301272963333119371, 5.77868922247834442174551932415, 5.78057953225196759515243928198, 5.92388708571680527569796532004, 6.57083924130414393763449692854, 6.73505616909807984622477097375, 6.96528112629200188798302820996, 6.96874588769461866320379606334, 7.14431026890047472265958664964, 7.910688366249232273735482437063, 8.112271176030091713646032641211, 8.199201998403981279907580734221, 8.561676929671817800086832661022, 8.716570320669932295560392336736

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