Properties

Degree $8$
Conductor $1.680\times 10^{14}$
Sign $1$
Motivic weight $0$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 16-s − 4·19-s − 4·49-s − 4·61-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + ⋯
L(s)  = 1  − 16-s − 4·19-s − 4·49-s − 4·61-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{8}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{3600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1897071349\)
\(L(\frac12)\) \(\approx\) \(0.1897071349\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
3 \( 1 \)
5 \( 1 \)
good7$C_2$ \( ( 1 + T^{2} )^{4} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_2$ \( ( 1 + T^{2} )^{4} \)
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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

−6.24597591718127930628764576941, −6.10727422809373159310536179858, −6.00477529917657942880727165907, −5.87879829589252799161627287048, −5.33182694746686031333768168516, −5.17550548795675326349954246620, −4.98217247665878702418684643061, −4.64682768377721411346024987088, −4.61350983078824726940189166123, −4.57505655808519907944404620231, −4.24353879066118819682560675171, −4.14603874487716110791227026129, −3.78500489946826328544568450396, −3.49023799246816021736920406591, −3.44094805437287590450927038824, −2.98954593424674399346939061585, −2.96059460854360747427595305849, −2.60723553073464479934983299064, −2.30813523539259161066549742251, −2.11342649451809806666003523781, −1.81600961662093426441045057405, −1.61437694517079147150296135734, −1.56873612633801131118794968377, −0.864490264831248719013907376912, −0.16671518267732937658998879965, 0.16671518267732937658998879965, 0.864490264831248719013907376912, 1.56873612633801131118794968377, 1.61437694517079147150296135734, 1.81600961662093426441045057405, 2.11342649451809806666003523781, 2.30813523539259161066549742251, 2.60723553073464479934983299064, 2.96059460854360747427595305849, 2.98954593424674399346939061585, 3.44094805437287590450927038824, 3.49023799246816021736920406591, 3.78500489946826328544568450396, 4.14603874487716110791227026129, 4.24353879066118819682560675171, 4.57505655808519907944404620231, 4.61350983078824726940189166123, 4.64682768377721411346024987088, 4.98217247665878702418684643061, 5.17550548795675326349954246620, 5.33182694746686031333768168516, 5.87879829589252799161627287048, 6.00477529917657942880727165907, 6.10727422809373159310536179858, 6.24597591718127930628764576941

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