Properties

Label 8-342e4-1.1-c1e4-0-1
Degree $8$
Conductor $13680577296$
Sign $1$
Analytic cond. $55.6176$
Root an. cond. $1.65253$
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  + 2·2-s + 4-s − 4·7-s − 2·8-s + 6·13-s − 8·14-s − 4·16-s − 12·17-s + 16·19-s − 8·25-s + 12·26-s − 4·28-s − 2·32-s − 24·34-s + 32·38-s + 2·43-s + 12·47-s − 6·49-s − 16·50-s + 6·52-s + 8·56-s − 24·59-s + 2·61-s + 3·64-s − 6·67-s − 12·68-s − 12·71-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s − 1.51·7-s − 0.707·8-s + 1.66·13-s − 2.13·14-s − 16-s − 2.91·17-s + 3.67·19-s − 8/5·25-s + 2.35·26-s − 0.755·28-s − 0.353·32-s − 4.11·34-s + 5.19·38-s + 0.304·43-s + 1.75·47-s − 6/7·49-s − 2.26·50-s + 0.832·52-s + 1.06·56-s − 3.12·59-s + 0.256·61-s + 3/8·64-s − 0.733·67-s − 1.45·68-s − 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(55.6176\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{342} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.171376593\)
\(L(\frac12)\) \(\approx\) \(2.171376593\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
good5$C_2^3$ \( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \)
7$D_{4}$ \( ( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 4 T^{2} - 138 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 - 6 T + 23 T^{2} - 66 T^{3} + 108 T^{4} - 66 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 + 12 T + 92 T^{2} + 528 T^{3} + 2463 T^{4} + 528 p T^{5} + 92 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^3$ \( 1 + 14 T^{2} - 333 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^3$ \( 1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 34 T^{2} + 267 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 106 T^{2} + 5331 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 2 T - 59 T^{2} + 46 T^{3} + 1948 T^{4} + 46 p T^{5} - 59 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 12 T + 152 T^{2} - 1248 T^{3} + 10863 T^{4} - 1248 p T^{5} + 152 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 82 T^{2} + 3915 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 24 T + 320 T^{2} + 3312 T^{3} + 28071 T^{4} + 3312 p T^{5} + 320 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 2 T - 113 T^{2} + 10 T^{3} + 9724 T^{4} + 10 p T^{5} - 113 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 6 T + 77 T^{2} + 390 T^{3} + 540 T^{4} + 390 p T^{5} + 77 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 14 T + 25 T^{2} - 350 T^{3} + 9604 T^{4} - 350 p T^{5} + 25 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 18 T + 275 T^{2} - 3006 T^{3} + 30180 T^{4} - 3006 p T^{5} + 275 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 112 T^{2} + 16050 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^3$ \( 1 - 28 T^{2} - 7137 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 36 T + 662 T^{2} + 8280 T^{3} + 85395 T^{4} + 8280 p T^{5} + 662 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \)
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

−8.651232523334635659386595581057, −7.88515924101842551361510319302, −7.77621372555888151194854533956, −7.71289871846927027415641561069, −7.15047969610458823299614430336, −7.13201052171745851246307111779, −6.70121054443875997899314226758, −6.34726048780037001003210773892, −6.19128663714793441561582383377, −6.14078170313540040532124208569, −5.83233843612459099147996027573, −5.53240496602713295945333750228, −4.99147247261258788581948349198, −4.97331361303756280787554931794, −4.79520365470205555239280520475, −4.13722927605505089241139764573, −3.93525799524286160302387512794, −3.68481166797119784202498181439, −3.65394358649041158921829057325, −2.95640398964277002254706102000, −2.84714669519753786965368297361, −2.67404515563176692415438687557, −1.78508797749065650417287367966, −1.47386393601436772054816722080, −0.49795736631894980176576189003, 0.49795736631894980176576189003, 1.47386393601436772054816722080, 1.78508797749065650417287367966, 2.67404515563176692415438687557, 2.84714669519753786965368297361, 2.95640398964277002254706102000, 3.65394358649041158921829057325, 3.68481166797119784202498181439, 3.93525799524286160302387512794, 4.13722927605505089241139764573, 4.79520365470205555239280520475, 4.97331361303756280787554931794, 4.99147247261258788581948349198, 5.53240496602713295945333750228, 5.83233843612459099147996027573, 6.14078170313540040532124208569, 6.19128663714793441561582383377, 6.34726048780037001003210773892, 6.70121054443875997899314226758, 7.13201052171745851246307111779, 7.15047969610458823299614430336, 7.71289871846927027415641561069, 7.77621372555888151194854533956, 7.88515924101842551361510319302, 8.651232523334635659386595581057

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