Properties

Label 8-3072e4-1.1-c1e4-0-5
Degree $8$
Conductor $8.906\times 10^{13}$
Sign $1$
Analytic cond. $362070.$
Root an. cond. $4.95278$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 4·5-s − 4·7-s + 10·9-s + 8·13-s − 16·15-s + 16·21-s − 20·27-s + 12·29-s − 12·31-s − 16·35-s + 16·37-s − 32·39-s + 40·45-s − 4·49-s + 20·53-s + 16·61-s − 40·63-s + 32·65-s + 16·67-s + 8·71-s − 8·73-s − 12·79-s + 35·81-s − 48·87-s − 8·89-s − 32·91-s + ⋯
L(s)  = 1  − 2.30·3-s + 1.78·5-s − 1.51·7-s + 10/3·9-s + 2.21·13-s − 4.13·15-s + 3.49·21-s − 3.84·27-s + 2.22·29-s − 2.15·31-s − 2.70·35-s + 2.63·37-s − 5.12·39-s + 5.96·45-s − 4/7·49-s + 2.74·53-s + 2.04·61-s − 5.03·63-s + 3.96·65-s + 1.95·67-s + 0.949·71-s − 0.936·73-s − 1.35·79-s + 35/9·81-s − 5.14·87-s − 0.847·89-s − 3.35·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{40} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(362070.\)
Root analytic conductor: \(4.95278\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3072} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{40} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.865840018\)
\(L(\frac12)\) \(\approx\) \(2.865840018\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
good5$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 16 T^{2} - 44 T^{3} + 118 T^{4} - 44 p T^{5} + 16 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 20 T^{2} + 60 T^{3} + 186 T^{4} + 60 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 20 T^{2} - 32 T^{3} + 230 T^{4} - 32 p T^{5} + 20 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 56 T^{2} - 264 T^{3} + 1122 T^{4} - 264 p T^{5} + 56 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 36 T^{2} - 64 T^{3} + 662 T^{4} - 64 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 44 T^{2} - 64 T^{3} + 966 T^{4} - 64 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 144 T^{2} - 964 T^{3} + 6422 T^{4} - 964 p T^{5} + 144 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 164 T^{2} + 1140 T^{3} + 8218 T^{4} + 1140 p T^{5} + 164 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 200 T^{2} - 1552 T^{3} + 11010 T^{4} - 1552 p T^{5} + 200 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 100 T^{2} + 192 T^{3} + 4726 T^{4} + 192 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 76 T^{2} - 256 T^{3} + 2726 T^{4} - 256 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 336 T^{2} - 3452 T^{3} + 30134 T^{4} - 3452 p T^{5} + 336 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 296 T^{2} - 2704 T^{3} + 27618 T^{4} - 2704 p T^{5} + 296 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 4 p T^{2} - 2960 T^{3} + 27190 T^{4} - 2960 p T^{5} + 4 p^{3} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 252 T^{2} - 1512 T^{3} + 25766 T^{4} - 1512 p T^{5} + 252 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 196 T^{2} + 1816 T^{3} + 18022 T^{4} + 1816 p T^{5} + 196 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 148 T^{2} - 44 T^{3} + 794 T^{4} - 44 p T^{5} + 148 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 116 T^{2} - 160 T^{3} + 5510 T^{4} - 160 p T^{5} + 116 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 156 T^{2} + 504 T^{3} + 10022 T^{4} + 504 p T^{5} + 156 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 164 T^{2} + 768 T^{3} + 13510 T^{4} + 768 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.19148592311634122342441078321, −5.87907232179393807089773081520, −5.87210147855211633331650705514, −5.59779024063240824485900491089, −5.45945467550082185054211361824, −5.35130175166643358956935929894, −5.19114080822899924705615454903, −4.83147318091589642998273263721, −4.63598009374082431147807944454, −4.07400509126686743956747739581, −4.04897279529944352095370893819, −4.02891165608713847351015565897, −4.02043268620955719180630533188, −3.40283064774786394980810778129, −3.08613104818524905276246437440, −3.07501098781709785572695988197, −2.77902034394668592938736735592, −2.19585836955146979442182775257, −2.18129231152720039535627561190, −1.80983909119701770521804643619, −1.69524908028366097373143870760, −1.16694361824026132816022090914, −0.822924600345026823605653674730, −0.78780356777455354705024187437, −0.38558982328734312295465831128, 0.38558982328734312295465831128, 0.78780356777455354705024187437, 0.822924600345026823605653674730, 1.16694361824026132816022090914, 1.69524908028366097373143870760, 1.80983909119701770521804643619, 2.18129231152720039535627561190, 2.19585836955146979442182775257, 2.77902034394668592938736735592, 3.07501098781709785572695988197, 3.08613104818524905276246437440, 3.40283064774786394980810778129, 4.02043268620955719180630533188, 4.02891165608713847351015565897, 4.04897279529944352095370893819, 4.07400509126686743956747739581, 4.63598009374082431147807944454, 4.83147318091589642998273263721, 5.19114080822899924705615454903, 5.35130175166643358956935929894, 5.45945467550082185054211361824, 5.59779024063240824485900491089, 5.87210147855211633331650705514, 5.87907232179393807089773081520, 6.19148592311634122342441078321

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