Properties

Label 8-1900e4-1.1-c2e4-0-2
Degree $8$
Conductor $1.303\times 10^{13}$
Sign $1$
Analytic cond. $7.18380\times 10^{6}$
Root an. cond. $7.19522$
Motivic weight $2$
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  + 22·9-s + 56·11-s − 40·19-s + 194·49-s − 160·61-s + 201·81-s + 1.23e3·99-s + 56·101-s + 1.47e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 154·169-s − 880·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 22/9·9-s + 5.09·11-s − 2.10·19-s + 3.95·49-s − 2.62·61-s + 2.48·81-s + 12.4·99-s + 0.554·101-s + 12.1·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.911·169-s − 5.14·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(7.18380\times 10^{6}\)
Root analytic conductor: \(7.19522\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{8} \cdot 19^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(6.281463269\)
\(L(\frac12)\) \(\approx\) \(6.281463269\)
\(L(2)\) 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 \( 1 \)
5 \( 1 \)
19$C_2$ \( ( 1 + 20 T + p^{2} T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 - 11 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 97 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )^{4} \)
13$C_2^2$ \( ( 1 + 77 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 49 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 1057 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 23 p T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 878 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 1694 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 2318 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 926 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 3742 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 907 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 6701 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 40 T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 8717 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 9038 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 10609 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 3086 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 12754 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 862 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 9422 T^{2} + p^{4} T^{4} )^{2} \)
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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

−6.47077389979552894102218145931, −6.11050817206310978634847086898, −6.02027850387531600604936152991, −6.01838119390664391761688344969, −5.50418665006538476514017161693, −5.47809215338223679363054140363, −4.82387231719546786898293551993, −4.57992680007553200580072990031, −4.57855202345696301591488072292, −4.34227618281655049346286685220, −4.18191143335617995355826567517, −3.96051320713687802834343340740, −3.80694714752311074483267923994, −3.75162765688387679066940260592, −3.33874858115252401988410218141, −3.17348118886089477177487667295, −2.64853105100094016125835197321, −2.10587540819028764995550021786, −2.05178215671201800518529920714, −2.01987214720837216679695337451, −1.29443436105305469586945536014, −1.22485937892824282571502573794, −1.18865511936312330932264021290, −0.972324984351723281175123201606, −0.22925894315651974448596203637, 0.22925894315651974448596203637, 0.972324984351723281175123201606, 1.18865511936312330932264021290, 1.22485937892824282571502573794, 1.29443436105305469586945536014, 2.01987214720837216679695337451, 2.05178215671201800518529920714, 2.10587540819028764995550021786, 2.64853105100094016125835197321, 3.17348118886089477177487667295, 3.33874858115252401988410218141, 3.75162765688387679066940260592, 3.80694714752311074483267923994, 3.96051320713687802834343340740, 4.18191143335617995355826567517, 4.34227618281655049346286685220, 4.57855202345696301591488072292, 4.57992680007553200580072990031, 4.82387231719546786898293551993, 5.47809215338223679363054140363, 5.50418665006538476514017161693, 6.01838119390664391761688344969, 6.02027850387531600604936152991, 6.11050817206310978634847086898, 6.47077389979552894102218145931

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