Properties

Label 8-33e8-1.1-c2e4-0-4
Degree $8$
Conductor $1.406\times 10^{12}$
Sign $1$
Analytic cond. $775267.$
Root an. cond. $5.44730$
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  − 6·4-s − 5·16-s + 76·25-s − 176·31-s + 176·37-s + 180·64-s − 352·67-s + 280·97-s − 456·100-s + 56·103-s + 1.05e3·124-s + 127-s + 131-s + 137-s + 139-s − 1.05e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 352·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 3/2·4-s − 0.312·16-s + 3.03·25-s − 5.67·31-s + 4.75·37-s + 2.81·64-s − 5.25·67-s + 2.88·97-s − 4.55·100-s + 0.543·103-s + 8.51·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 7.13·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.08·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(775267.\)
Root analytic conductor: \(5.44730\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1089} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 11^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.803478748\)
\(L(\frac12)\) \(\approx\) \(1.803478748\)
\(L(2)\) 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)$
bad3 \( 1 \)
11 \( 1 \)
good2$C_2^2$ \( ( 1 + 3 T^{2} + p^{4} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 - 38 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 176 T^{2} + p^{4} T^{4} )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
19$C_2^2$ \( ( 1 - 704 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 266 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 1506 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 44 T + p^{2} T^{2} )^{4} \)
37$C_2$ \( ( 1 - 44 T + p^{2} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 1778 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 2816 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 4330 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 1306 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 6610 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 2640 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2$ \( ( 1 + 88 T + p^{2} T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + 7882 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 8976 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 12320 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 5154 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 3170 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 70 T + p^{2} T^{2} )^{4} \)
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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

−6.97534700776859454940076125510, −6.56920063318945037853820620513, −6.54277218688418515081979300022, −6.00977486030524172566782069196, −5.99868530817628027886733231835, −5.58698683906488701723096275243, −5.51852723810499299525545315646, −5.37003515282712208716911551540, −4.81728474488088387969969853155, −4.70550019212804737248923362077, −4.64814669274223822233158574322, −4.37425898827163000776343733820, −4.13745357914872663926915114863, −3.89698079789870994027527128323, −3.51089818105433550452126976807, −3.41393027928313446351803696745, −2.86673208673362152082037337584, −2.76578377136109493261548190997, −2.61983034328304331452748967319, −1.92653863611851370656441160269, −1.72826571302154850162004140093, −1.56903068604804080264196699389, −0.74561151750128686202823466109, −0.67601336902735647994752421435, −0.29555585704730719186027147404, 0.29555585704730719186027147404, 0.67601336902735647994752421435, 0.74561151750128686202823466109, 1.56903068604804080264196699389, 1.72826571302154850162004140093, 1.92653863611851370656441160269, 2.61983034328304331452748967319, 2.76578377136109493261548190997, 2.86673208673362152082037337584, 3.41393027928313446351803696745, 3.51089818105433550452126976807, 3.89698079789870994027527128323, 4.13745357914872663926915114863, 4.37425898827163000776343733820, 4.64814669274223822233158574322, 4.70550019212804737248923362077, 4.81728474488088387969969853155, 5.37003515282712208716911551540, 5.51852723810499299525545315646, 5.58698683906488701723096275243, 5.99868530817628027886733231835, 6.00977486030524172566782069196, 6.54277218688418515081979300022, 6.56920063318945037853820620513, 6.97534700776859454940076125510

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