Properties

Degree $8$
Conductor $1358954496$
Sign $1$
Motivic weight $2$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·9-s + 24·13-s + 84·25-s + 152·37-s + 20·49-s + 88·61-s − 120·73-s − 45·81-s + 360·97-s − 552·109-s − 144·117-s + 52·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 316·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2/3·9-s + 1.84·13-s + 3.35·25-s + 4.10·37-s + 0.408·49-s + 1.44·61-s − 1.64·73-s − 5/9·81-s + 3.71·97-s − 5.06·109-s − 1.23·117-s + 0.429·121-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 − 1.86·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{4}\)
Sign: $1$
Motivic weight: \(2\)
Character: induced by $\chi_{192} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.64300\)
\(L(\frac12)\) \(\approx\) \(3.64300\)
\(L(2)\) 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 \( 1 \)
3$C_2^2$ \( 1 + 2 p T^{2} + p^{4} T^{4} \)
good5$C_2^2$ \( ( 1 - 42 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 10 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 26 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p^{2} T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 66 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 614 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 194 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 714 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 950 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 38 T + p^{2} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 3330 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 3590 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 962 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 5418 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 6746 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 22 T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 4090 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 9218 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 30 T + p^{2} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 11510 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 8378 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 15810 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 90 T + p^{2} T^{2} )^{4} \)
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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

−8.890019075588946499470974626715, −8.692018812970710695813805880800, −8.535448451387586839593442363822, −8.189674277927554316364104022505, −7.954762290277353143503316656915, −7.45436655754535338171488764119, −7.44175615020804701800707046341, −7.02679106083668334129796750170, −6.55217722705508720723459399003, −6.48029169248835227040527329569, −6.09999779994752604948496080480, −5.84481953356596687067487609140, −5.80458253610932666629022668597, −4.99712288124420771931009101710, −4.86792930215584232727671835704, −4.81896274423511660699511031808, −4.00362511082225894223277908158, −3.92772572483879986084655211127, −3.63447412582824223831344670747, −2.80087441125730779139323786691, −2.75157826694989290057115602843, −2.63146738052183941462554433015, −1.57108700097724275686614189191, −1.06871512151281810989255269874, −0.72401904427738249545463080342, 0.72401904427738249545463080342, 1.06871512151281810989255269874, 1.57108700097724275686614189191, 2.63146738052183941462554433015, 2.75157826694989290057115602843, 2.80087441125730779139323786691, 3.63447412582824223831344670747, 3.92772572483879986084655211127, 4.00362511082225894223277908158, 4.81896274423511660699511031808, 4.86792930215584232727671835704, 4.99712288124420771931009101710, 5.80458253610932666629022668597, 5.84481953356596687067487609140, 6.09999779994752604948496080480, 6.48029169248835227040527329569, 6.55217722705508720723459399003, 7.02679106083668334129796750170, 7.44175615020804701800707046341, 7.45436655754535338171488764119, 7.954762290277353143503316656915, 8.189674277927554316364104022505, 8.535448451387586839593442363822, 8.692018812970710695813805880800, 8.890019075588946499470974626715

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