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Properties

Label 8-4004e4-1.1-c1e4-0-0
Degree $8$
Conductor $2.570\times 10^{14}$
Sign $1$
Analytic cond. $1.04492\times 10^{6}$
Root an. cond. $5.65438$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 4·7-s − 7·9-s + 4·11-s − 4·13-s + 15-s + 17-s − 3·19-s − 4·21-s − 23-s − 15·25-s − 10·27-s + 3·29-s + 6·31-s + 4·33-s − 4·35-s + 4·37-s − 4·39-s − 6·41-s − 3·43-s − 7·45-s − 7·47-s + 10·49-s + 51-s − 20·53-s + 4·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.51·7-s − 7/3·9-s + 1.20·11-s − 1.10·13-s + 0.258·15-s + 0.242·17-s − 0.688·19-s − 0.872·21-s − 0.208·23-s − 3·25-s − 1.92·27-s + 0.557·29-s + 1.07·31-s + 0.696·33-s − 0.676·35-s + 0.657·37-s − 0.640·39-s − 0.937·41-s − 0.457·43-s − 1.04·45-s − 1.02·47-s + 10/7·49-s + 0.140·51-s − 2.74·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 7^{4} \cdot 11^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1.04492\times 10^{6}\)
Root analytic conductor: \(5.65438\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{8} \cdot 7^{4} \cdot 11^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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 \( 1 \)
7$C_1$ \( ( 1 + T )^{4} \)
11$C_1$ \( ( 1 - T )^{4} \)
13$C_1$ \( ( 1 + T )^{4} \)
good3$((C_8 : C_2):C_2):C_2$ \( 1 - T + 8 T^{2} - 5 T^{3} + 31 T^{4} - 5 p T^{5} + 8 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
5$((C_8 : C_2):C_2):C_2$ \( 1 - T + 16 T^{2} - 11 T^{3} + 111 T^{4} - 11 p T^{5} + 16 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 - T + 44 T^{2} + 23 T^{3} + 859 T^{4} + 23 p T^{5} + 44 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 + 3 T + 60 T^{2} + 153 T^{3} + 1589 T^{4} + 153 p T^{5} + 60 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 + T + 88 T^{2} + 65 T^{3} + 2991 T^{4} + 65 p T^{5} + 88 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 - 3 T + 80 T^{2} - 153 T^{3} + 3039 T^{4} - 153 p T^{5} + 80 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 - 6 T + 120 T^{2} - 504 T^{3} + 5489 T^{4} - 504 p T^{5} + 120 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 94 T^{2} - 373 T^{3} + 4249 T^{4} - 373 p T^{5} + 94 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 + 6 T + 100 T^{2} + 249 T^{3} + 4179 T^{4} + 249 p T^{5} + 100 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 + 3 T + 141 T^{2} + 384 T^{3} + 8429 T^{4} + 384 p T^{5} + 141 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 + 7 T + 142 T^{2} + 785 T^{3} + 9741 T^{4} + 785 p T^{5} + 142 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 + 20 T + 247 T^{2} + 1870 T^{3} + 13959 T^{4} + 1870 p T^{5} + 247 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 + 11 T + 127 T^{2} + 1078 T^{3} + 10335 T^{4} + 1078 p T^{5} + 127 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 + 18 T + 318 T^{2} + 3291 T^{3} + 31325 T^{4} + 3291 p T^{5} + 318 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 + 16 T + 294 T^{2} + 3032 T^{3} + 30479 T^{4} + 3032 p T^{5} + 294 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 178 T^{2} - 1135 T^{3} + 15751 T^{4} - 1135 p T^{5} + 178 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 - 9 T + 247 T^{2} - 1242 T^{3} + 24375 T^{4} - 1242 p T^{5} + 247 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 + 14 T + 358 T^{2} + 3385 T^{3} + 45681 T^{4} + 3385 p T^{5} + 358 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 + 23 T + 410 T^{2} + 4313 T^{3} + 46459 T^{4} + 4313 p T^{5} + 410 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 - 14 T + 204 T^{2} - 2593 T^{3} + 32969 T^{4} - 2593 p T^{5} + 204 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.37920073912689053281199461394, −6.11422259107529550877153007796, −6.00263703759491526335786276763, −5.95005298285687925639825661631, −5.76797843694287366093188027295, −5.38441567822445799364782407921, −5.14503444369706779597684160915, −5.03824637496173388855554940191, −4.89090177087398263760832320003, −4.43525450851473754373342687096, −4.23072700382223079836904930523, −4.20476946006541624934331093185, −4.06513182784230762222586423799, −3.49227547180444923993031098197, −3.31298396931668805225902990335, −3.29323498573313191794390188494, −3.26298378967128612215102095874, −2.77402988893210592892743056964, −2.53392617876398331635828134139, −2.47490101735427255017314982929, −2.40877138875909727999203600878, −1.79402882292903867678071108436, −1.53620006246135482142359615155, −1.45096949731917424460525107782, −1.13940660503600657452161457975, 0, 0, 0, 0, 1.13940660503600657452161457975, 1.45096949731917424460525107782, 1.53620006246135482142359615155, 1.79402882292903867678071108436, 2.40877138875909727999203600878, 2.47490101735427255017314982929, 2.53392617876398331635828134139, 2.77402988893210592892743056964, 3.26298378967128612215102095874, 3.29323498573313191794390188494, 3.31298396931668805225902990335, 3.49227547180444923993031098197, 4.06513182784230762222586423799, 4.20476946006541624934331093185, 4.23072700382223079836904930523, 4.43525450851473754373342687096, 4.89090177087398263760832320003, 5.03824637496173388855554940191, 5.14503444369706779597684160915, 5.38441567822445799364782407921, 5.76797843694287366093188027295, 5.95005298285687925639825661631, 6.00263703759491526335786276763, 6.11422259107529550877153007796, 6.37920073912689053281199461394

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