↑

Properties

Label 8-7500e4-1.1-c1e4-0-7
Degree $8$
Conductor $3.164\times 10^{15}$
Sign $1$
Analytic cond. $1.28633\times 10^{7}$
Root an. cond. $7.73872$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 4·7-s + 10·9-s − 11-s + 5·13-s − 4·17-s − 5·19-s − 16·21-s − 9·23-s + 20·27-s − 4·29-s − 9·31-s − 4·33-s − 2·37-s + 20·39-s − 34·43-s − 6·47-s − 8·49-s − 16·51-s + 2·53-s − 20·57-s + 59-s − 2·61-s − 40·63-s + 4·67-s − 36·69-s − 20·71-s + ⋯
L(s)  = 1  + 2.30·3-s − 1.51·7-s + 10/3·9-s − 0.301·11-s + 1.38·13-s − 0.970·17-s − 1.14·19-s − 3.49·21-s − 1.87·23-s + 3.84·27-s − 0.742·29-s − 1.61·31-s − 0.696·33-s − 0.328·37-s + 3.20·39-s − 5.18·43-s − 0.875·47-s − 8/7·49-s − 2.24·51-s + 0.274·53-s − 2.64·57-s + 0.130·59-s − 0.256·61-s − 5.03·63-s + 0.488·67-s − 4.33·69-s − 2.37·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{4} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(1.28633\times 10^{7}\)
Root analytic conductor: \(7.73872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{8} \cdot 3^{4} \cdot 5^{16} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
5 \( 1 \)
good7$((C_8 : C_2):C_2):C_2$ \( 1 + 4 T + 24 T^{2} + 83 T^{3} + 239 T^{4} + 83 p T^{5} + 24 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
11$((C_8 : C_2):C_2):C_2$ \( 1 + T + 40 T^{2} + 29 T^{3} + 639 T^{4} + 29 p T^{5} + 40 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
13$((C_8 : C_2):C_2):C_2$ \( 1 - 5 T + 42 T^{2} - 175 T^{3} + 749 T^{4} - 175 p T^{5} + 42 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 + 4 T + 2 p T^{2} + 143 T^{3} + 849 T^{4} + 143 p T^{5} + 2 p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 + 5 T + 51 T^{2} + 10 p T^{3} + 1361 T^{4} + 10 p^{2} T^{5} + 51 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 + 9 T + 88 T^{2} + 495 T^{3} + 3021 T^{4} + 495 p T^{5} + 88 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 + 4 T + 22 T^{2} - 223 T^{3} - 1005 T^{4} - 223 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 9 T + 130 T^{2} + 711 T^{3} + 189 p T^{4} + 711 p T^{5} + 130 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 102 T^{2} + 205 T^{3} + 4991 T^{4} + 205 p T^{5} + 102 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 + 94 T^{2} + 240 T^{3} + 4191 T^{4} + 240 p T^{5} + 94 p^{2} T^{6} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 + 34 T + 573 T^{2} + 6230 T^{3} + 47951 T^{4} + 6230 p T^{5} + 573 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 + 6 T + 2 p T^{2} + 507 T^{3} + 6039 T^{4} + 507 p T^{5} + 2 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 166 T^{2} - 121 T^{3} + 11799 T^{4} - 121 p T^{5} + 166 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 - T + 97 T^{2} - 218 T^{3} + 6825 T^{4} - 218 p T^{5} + 97 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 148 T^{2} + 764 T^{3} + 10165 T^{4} + 764 p T^{5} + 148 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 109 T^{2} + 512 T^{3} + 3169 T^{4} + 512 p T^{5} + 109 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 + 20 T + 334 T^{2} + 3385 T^{3} + 33471 T^{4} + 3385 p T^{5} + 334 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 + 12 T + 151 T^{2} + 1566 T^{3} + 9759 T^{4} + 1566 p T^{5} + 151 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 + 3 T + 130 T^{2} - 567 T^{3} + 6699 T^{4} - 567 p T^{5} + 130 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 + 14 T + 388 T^{2} + 3520 T^{3} + 50541 T^{4} + 3520 p T^{5} + 388 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 - 15 T + 386 T^{2} - 3915 T^{3} + 52911 T^{4} - 3915 p T^{5} + 386 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 + 28 T + 6 p T^{2} + 7880 T^{3} + 89831 T^{4} + 7880 p T^{5} + 6 p^{3} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
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

−5.95118749626621891048582043515, −5.73934239659270620898062875518, −5.62108297755608466676270751184, −5.25034016916104295939772364538, −5.24464202090665968031277015114, −4.81887921756291012121314941907, −4.69214840936118128922981503160, −4.51390620716124466637670962789, −4.44813906607628822212205590784, −3.96078308231019895119552553792, −3.83230806007196359795981890729, −3.77542800263934235806130595144, −3.71370611834180232909411823144, −3.30282614106123155614133818878, −3.26228533073044497026202779473, −3.10508360442687710480401431142, −3.02673707695769526978634943027, −2.50091655989556031137545569332, −2.30440286563450660839587159180, −2.18639411239472032429066874157, −2.17654610111489065724977012175, −1.49525599362996742127685520821, −1.48268923219602120544260153092, −1.45663999495648893889550801483, −1.27547642180696603633629781565, 0, 0, 0, 0, 1.27547642180696603633629781565, 1.45663999495648893889550801483, 1.48268923219602120544260153092, 1.49525599362996742127685520821, 2.17654610111489065724977012175, 2.18639411239472032429066874157, 2.30440286563450660839587159180, 2.50091655989556031137545569332, 3.02673707695769526978634943027, 3.10508360442687710480401431142, 3.26228533073044497026202779473, 3.30282614106123155614133818878, 3.71370611834180232909411823144, 3.77542800263934235806130595144, 3.83230806007196359795981890729, 3.96078308231019895119552553792, 4.44813906607628822212205590784, 4.51390620716124466637670962789, 4.69214840936118128922981503160, 4.81887921756291012121314941907, 5.24464202090665968031277015114, 5.25034016916104295939772364538, 5.62108297755608466676270751184, 5.73934239659270620898062875518, 5.95118749626621891048582043515

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