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Properties

Label 8-7616e4-1.1-c1e4-0-7
Degree $8$
Conductor $3.364\times 10^{15}$
Sign $1$
Analytic cond. $1.36777\times 10^{7}$
Root an. cond. $7.79833$
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  + 4·5-s + 4·7-s − 7·9-s − 2·11-s + 2·13-s + 4·17-s − 4·19-s − 4·23-s − 5·25-s − 6·29-s − 16·31-s + 16·35-s + 8·37-s − 10·41-s − 4·43-s − 28·45-s − 14·47-s + 10·49-s + 14·53-s − 8·55-s − 12·59-s + 2·61-s − 28·63-s + 8·65-s − 22·67-s − 16·71-s − 14·73-s + ⋯
L(s)  = 1  + 1.78·5-s + 1.51·7-s − 7/3·9-s − 0.603·11-s + 0.554·13-s + 0.970·17-s − 0.917·19-s − 0.834·23-s − 25-s − 1.11·29-s − 2.87·31-s + 2.70·35-s + 1.31·37-s − 1.56·41-s − 0.609·43-s − 4.17·45-s − 2.04·47-s + 10/7·49-s + 1.92·53-s − 1.07·55-s − 1.56·59-s + 0.256·61-s − 3.52·63-s + 0.992·65-s − 2.68·67-s − 1.89·71-s − 1.63·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 7^{4} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(1.36777\times 10^{7}\)
Root analytic conductor: \(7.79833\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{24} \cdot 7^{4} \cdot 17^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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$C_1$ \( ( 1 - T )^{4} \)
17$C_1$ \( ( 1 - T )^{4} \)
good3$C_2^2:C_4$ \( 1 + 7 T^{2} + 29 T^{4} + 7 p^{2} T^{6} + p^{4} T^{8} \)
5$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 21 T^{2} - 54 T^{3} + 161 T^{4} - 54 p T^{5} + 21 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
11$((C_8 : C_2):C_2):C_2$ \( 1 + 2 T + 38 T^{2} + 54 T^{3} + 590 T^{4} + 54 p T^{5} + 38 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 2 p T^{2} - 66 T^{3} + 414 T^{4} - 66 p T^{5} + 2 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 + 4 T + 32 T^{2} + 132 T^{3} + 1050 T^{4} + 132 p T^{5} + 32 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 + 4 T + 88 T^{2} + 260 T^{3} + 2986 T^{4} + 260 p T^{5} + 88 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 + 6 T + 2 T^{2} + 118 T^{3} + 1550 T^{4} + 118 p T^{5} + 2 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 16 T + 175 T^{2} + 44 p T^{3} + 8689 T^{4} + 44 p^{2} T^{5} + 175 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 - 8 T + 112 T^{2} - 480 T^{3} + 4826 T^{4} - 480 p T^{5} + 112 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 109 T^{2} + 650 T^{3} + 4881 T^{4} + 650 p T^{5} + 109 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 + 4 T + 93 T^{2} + 170 T^{3} + 3941 T^{4} + 170 p T^{5} + 93 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 + 14 T + 194 T^{2} + 1818 T^{3} + 13574 T^{4} + 1818 p T^{5} + 194 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 - 14 T + 243 T^{2} - 2040 T^{3} + 19541 T^{4} - 2040 p T^{5} + 243 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 + 12 T + 100 T^{2} + 492 T^{3} + 934 T^{4} + 492 p T^{5} + 100 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 - 2 T + 53 T^{2} - 74 T^{3} + 45 T^{4} - 74 p T^{5} + 53 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 + 22 T + 377 T^{2} + 4190 T^{3} + 40381 T^{4} + 4190 p T^{5} + 377 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 + 16 T + 320 T^{2} + 2984 T^{3} + 33754 T^{4} + 2984 p T^{5} + 320 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 + 14 T + 253 T^{2} + 2990 T^{3} + 26461 T^{4} + 2990 p T^{5} + 253 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 326 T^{2} + 2350 T^{3} + 39006 T^{4} + 2350 p T^{5} + 326 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 - 6 T + 138 T^{2} - 1050 T^{3} + 15366 T^{4} - 1050 p T^{5} + 138 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 + 22 T + 390 T^{2} + 4822 T^{3} + 54134 T^{4} + 4822 p T^{5} + 390 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 + 12 T + 217 T^{2} + 1090 T^{3} + 19061 T^{4} + 1090 p T^{5} + 217 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.77395585617724104144959375282, −5.60673611873507073765118700106, −5.56963743894999302515653257630, −5.53215172025410825481425463019, −5.29943375450906817058743800561, −5.18630049264230315156167608355, −4.81935913650205945355168559424, −4.69808786375592453126089706171, −4.37149824184694968333745610673, −4.24358712490388287300353561889, −4.03308541396408244723383671780, −3.78753337957725529609952522208, −3.73299719647478410358292166152, −3.29992350526106221147795794172, −3.09399106023424527039834708895, −3.04307197706858824747922970885, −2.88687067534941586952659281812, −2.37288673267512257399591681326, −2.23512973828654404504520326338, −2.17370969668862096679586464355, −2.04557025203933149161042945176, −1.59091357660204212844604942816, −1.41273727717312973504359349461, −1.29897629261242221516898546625, −1.28149416781536503721554254703, 0, 0, 0, 0, 1.28149416781536503721554254703, 1.29897629261242221516898546625, 1.41273727717312973504359349461, 1.59091357660204212844604942816, 2.04557025203933149161042945176, 2.17370969668862096679586464355, 2.23512973828654404504520326338, 2.37288673267512257399591681326, 2.88687067534941586952659281812, 3.04307197706858824747922970885, 3.09399106023424527039834708895, 3.29992350526106221147795794172, 3.73299719647478410358292166152, 3.78753337957725529609952522208, 4.03308541396408244723383671780, 4.24358712490388287300353561889, 4.37149824184694968333745610673, 4.69808786375592453126089706171, 4.81935913650205945355168559424, 5.18630049264230315156167608355, 5.29943375450906817058743800561, 5.53215172025410825481425463019, 5.56963743894999302515653257630, 5.60673611873507073765118700106, 5.77395585617724104144959375282

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