Properties

Label 8-1152e4-1.1-c2e4-0-3
Degree $8$
Conductor $1.761\times 10^{12}$
Sign $1$
Analytic cond. $970845.$
Root an. cond. $5.60265$
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  − 96·25-s − 196·49-s − 384·73-s − 576·97-s − 484·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 476·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 3.83·25-s − 4·49-s − 5.26·73-s − 5.93·97-s − 4·121-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 + 2.81·169-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 + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(970845.\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.07663744112\)
\(L(\frac12)\) \(\approx\) \(0.07663744112\)
\(L(2)\) 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 \( 1 \)
good5$C_2^2$ \( ( 1 + 48 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
11$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
13$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )^{2}( 1 + 24 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( ( 1 - 480 T^{2} + p^{4} T^{4} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
29$C_2^2$ \( ( 1 + 1680 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
37$C_2$ \( ( 1 - 70 T + p^{2} T^{2} )^{2}( 1 + 70 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( ( 1 - 1440 T^{2} + p^{4} T^{4} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
53$C_2^2$ \( ( 1 - 5040 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
61$C_2$ \( ( 1 - 22 T + p^{2} T^{2} )^{2}( 1 + 22 T + p^{2} T^{2} )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
73$C_2$ \( ( 1 + 96 T + p^{2} T^{2} )^{4} \)
79$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
83$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
89$C_2^2$ \( ( 1 - 12480 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 144 T + p^{2} T^{2} )^{4} \)
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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

−6.73409262861747162276703595520, −6.65956751128351391715288950964, −6.44132915302646410752280583769, −5.88362042695444301390147581064, −5.86025589245632557193356903111, −5.80169330441415293404076923427, −5.69679986082384336136700266461, −5.22926650391183677909339365195, −4.89493549461387179182266966862, −4.76472020809121461654195407482, −4.61146326919192269816293046015, −4.08984967877816730073850536097, −3.92487389608208081990526233590, −3.87691394928541556837476485420, −3.73110585597308865210701653239, −3.03732579121269054906311726144, −2.93502635702133438127901364309, −2.80288814735066651683582761967, −2.47221788160497823330499447943, −1.91588778574575743945702042982, −1.58805408817701538525592718773, −1.54808679605004454174135947935, −1.36438076980014877138084177606, −0.41873058858541769324935210586, −0.05832625287861352165766350484, 0.05832625287861352165766350484, 0.41873058858541769324935210586, 1.36438076980014877138084177606, 1.54808679605004454174135947935, 1.58805408817701538525592718773, 1.91588778574575743945702042982, 2.47221788160497823330499447943, 2.80288814735066651683582761967, 2.93502635702133438127901364309, 3.03732579121269054906311726144, 3.73110585597308865210701653239, 3.87691394928541556837476485420, 3.92487389608208081990526233590, 4.08984967877816730073850536097, 4.61146326919192269816293046015, 4.76472020809121461654195407482, 4.89493549461387179182266966862, 5.22926650391183677909339365195, 5.69679986082384336136700266461, 5.80169330441415293404076923427, 5.86025589245632557193356903111, 5.88362042695444301390147581064, 6.44132915302646410752280583769, 6.65956751128351391715288950964, 6.73409262861747162276703595520

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