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Properties

Label 8-7623e4-1.1-c1e4-0-2
Degree $8$
Conductor $3.377\times 10^{15}$
Sign $1$
Analytic cond. $1.37281\times 10^{7}$
Root an. cond. $7.80192$
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  − 2·2-s + 2·5-s − 4·7-s + 4·8-s − 4·10-s + 10·13-s + 8·14-s − 6·16-s − 6·17-s + 18·19-s + 2·23-s − 12·25-s − 20·26-s − 6·29-s + 12·34-s − 8·35-s − 4·37-s − 36·38-s + 8·40-s + 20·43-s − 4·46-s + 6·47-s + 10·49-s + 24·50-s − 16·56-s + 12·58-s + 6·59-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.894·5-s − 1.51·7-s + 1.41·8-s − 1.26·10-s + 2.77·13-s + 2.13·14-s − 3/2·16-s − 1.45·17-s + 4.12·19-s + 0.417·23-s − 2.39·25-s − 3.92·26-s − 1.11·29-s + 2.05·34-s − 1.35·35-s − 0.657·37-s − 5.83·38-s + 1.26·40-s + 3.04·43-s − 0.589·46-s + 0.875·47-s + 10/7·49-s + 3.39·50-s − 2.13·56-s + 1.57·58-s + 0.781·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 7^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(1.37281\times 10^{7}\)
Root analytic conductor: \(7.80192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.374556550\)
\(L(\frac12)\) \(\approx\) \(2.374556550\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 + p T + p^{2} T^{2} + p^{2} T^{3} + 3 p T^{4} + p^{3} T^{5} + p^{4} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 16 T^{2} - 28 T^{3} + 111 T^{4} - 28 p T^{5} + 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 84 T^{2} - 422 T^{3} + 1844 T^{4} - 422 p T^{5} + 84 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 28 T^{2} + 24 T^{3} + 111 T^{4} + 24 p T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 18 T + 184 T^{2} - 1278 T^{3} + 6468 T^{4} - 1278 p T^{5} + 184 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 28 T^{2} - 190 T^{3} + 516 T^{4} - 190 p T^{5} + 28 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 20 T^{2} + 126 T^{3} + 1404 T^{4} + 126 p T^{5} + 20 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 28 T^{2} - 216 T^{3} + 606 T^{4} - 216 p T^{5} + 28 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 72 T^{2} + 284 T^{3} + 4082 T^{4} + 284 p T^{5} + 72 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 124 T^{2} - 48 T^{3} + 6810 T^{4} - 48 p T^{5} + 124 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 282 T^{2} - 2632 T^{3} + 19943 T^{4} - 2632 p T^{5} + 282 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 44 T^{2} - 504 T^{3} + 93 p T^{4} - 504 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 148 T^{2} + 192 T^{3} + 9942 T^{4} + 192 p T^{5} + 148 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 208 T^{2} - 936 T^{3} + 17751 T^{4} - 936 p T^{5} + 208 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 108 T^{2} + 314 T^{3} + 3596 T^{4} + 314 p T^{5} + 108 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 130 T^{2} + 56 T^{3} + 8299 T^{4} + 56 p T^{5} + 130 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 220 T^{2} - 1014 T^{3} + 20916 T^{4} - 1014 p T^{5} + 220 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 34 T + 708 T^{2} - 9614 T^{3} + 96764 T^{4} - 9614 p T^{5} + 708 p^{2} T^{6} - 34 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 24 T + 280 T^{2} - 1584 T^{3} + 8754 T^{4} - 1584 p T^{5} + 280 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 260 T^{2} + 1188 T^{3} + 30039 T^{4} + 1188 p T^{5} + 260 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 18 T + 352 T^{2} + 4068 T^{3} + 48255 T^{4} + 4068 p T^{5} + 352 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 4 p T^{2} + 2758 T^{3} + 56428 T^{4} + 2758 p T^{5} + 4 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.70229463829404540165616980856, −5.28131912055786420394527595784, −5.17303800936425673162365874525, −5.14703909004477386609851093633, −5.11704220482709272249665947236, −4.41506514832853475501553889364, −4.26893044942699865281468193617, −4.10862188423722181494975232557, −4.05476586550508274327281657198, −3.75017270861013256891100094673, −3.54988574335104572415166593160, −3.51054556026176208468965434674, −3.39249599567391088803258740906, −2.91452236896463343536536320195, −2.83414605232112325262456236295, −2.39634956127631973810816390912, −2.38197287142304164349499752703, −2.16276951777660565528606488146, −1.68798279457400041827119586532, −1.46549675322359722394018249135, −1.42647708344282624659280736716, −1.03969336350816771079883236663, −0.794208413771655363229106371908, −0.48999723273947608286955237481, −0.36522644813463598699688246160, 0.36522644813463598699688246160, 0.48999723273947608286955237481, 0.794208413771655363229106371908, 1.03969336350816771079883236663, 1.42647708344282624659280736716, 1.46549675322359722394018249135, 1.68798279457400041827119586532, 2.16276951777660565528606488146, 2.38197287142304164349499752703, 2.39634956127631973810816390912, 2.83414605232112325262456236295, 2.91452236896463343536536320195, 3.39249599567391088803258740906, 3.51054556026176208468965434674, 3.54988574335104572415166593160, 3.75017270861013256891100094673, 4.05476586550508274327281657198, 4.10862188423722181494975232557, 4.26893044942699865281468193617, 4.41506514832853475501553889364, 5.11704220482709272249665947236, 5.14703909004477386609851093633, 5.17303800936425673162365874525, 5.28131912055786420394527595784, 5.70229463829404540165616980856

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