Properties

Degree $8$
Conductor $409600000000$
Sign $1$
Motivic weight $5$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 612·9-s − 1.64e4·29-s + 2.90e4·41-s + 2.87e4·49-s + 1.07e5·61-s + 1.62e5·81-s + 4.30e5·89-s + 2.36e5·101-s + 5.58e5·109-s − 5.09e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.43e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 2.51·9-s − 3.63·29-s + 2.70·41-s + 1.71·49-s + 3.68·61-s + 2.75·81-s + 5.75·89-s + 2.30·101-s + 4.50·109-s − 3.16·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 3.87·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 5^{8}\)
Sign: $1$
Motivic weight: \(5\)
Character: induced by $\chi_{800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 5^{8} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(19.44222330\)
\(L(\frac12)\) \(\approx\) \(19.44222330\)
\(L(\frac{7}{2})\) 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 \( 1 \)
good3$C_2^2$ \( ( 1 - 34 p^{2} T^{2} + p^{10} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 14394 T^{2} + p^{10} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 254822 T^{2} + p^{10} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 718870 T^{2} + p^{10} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 2808030 T^{2} + p^{10} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 4019078 T^{2} + p^{10} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 5934266 T^{2} + p^{10} T^{4} )^{2} \)
29$C_2$ \( ( 1 + 4110 T + p^{5} T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 47289582 T^{2} + p^{10} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 83304550 T^{2} + p^{10} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 7270 T + p^{5} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 26783614 T^{2} + p^{10} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 403777034 T^{2} + p^{10} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 202124090 T^{2} + p^{10} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 270997718 T^{2} + p^{10} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 26770 T + p^{5} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 219594834 T^{2} + p^{10} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 681226622 T^{2} + p^{10} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 3802634030 T^{2} + p^{10} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 1369983522 T^{2} + p^{10} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 1693436786 T^{2} + p^{10} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 107590 T + p^{5} T^{2} )^{4} \)
97$C_2^2$ \( ( 1 - 5328970270 T^{2} + p^{10} T^{4} )^{2} \)
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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

−6.65553626169735182784301470583, −6.24403911066181939868423854822, −6.07839447024359736275730452210, −6.06417016047626141266947412115, −5.70523311621998698547455519608, −5.35273911334548315379133226390, −5.09478022437639887587343039248, −4.91786188293768492943996423392, −4.88539348994232004145439305586, −4.18525394758068224106471851620, −4.06574541267715744060213379862, −3.99412607673623166964252787070, −3.94266725546524276282007294041, −3.52270301494093885297101614858, −3.29034309820472129584872968806, −2.74817825128681814274466501333, −2.64575071258979439869286728521, −2.03988482455286953805762988253, −1.88403841735864449719640416385, −1.82674862289652140693352811158, −1.72378999548811139924873476545, −0.847397695707074942104226766243, −0.71962466511199404234081476054, −0.71153054387486928716542038399, −0.46733847554135373672318054086, 0.46733847554135373672318054086, 0.71153054387486928716542038399, 0.71962466511199404234081476054, 0.847397695707074942104226766243, 1.72378999548811139924873476545, 1.82674862289652140693352811158, 1.88403841735864449719640416385, 2.03988482455286953805762988253, 2.64575071258979439869286728521, 2.74817825128681814274466501333, 3.29034309820472129584872968806, 3.52270301494093885297101614858, 3.94266725546524276282007294041, 3.99412607673623166964252787070, 4.06574541267715744060213379862, 4.18525394758068224106471851620, 4.88539348994232004145439305586, 4.91786188293768492943996423392, 5.09478022437639887587343039248, 5.35273911334548315379133226390, 5.70523311621998698547455519608, 6.06417016047626141266947412115, 6.07839447024359736275730452210, 6.24403911066181939868423854822, 6.65553626169735182784301470583

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