Properties

Label 8-1792e4-1.1-c1e4-0-9
Degree $8$
Conductor $1.031\times 10^{13}$
Sign $1$
Analytic cond. $41923.7$
Root an. cond. $3.78274$
Motivic weight $1$
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  + 8·25-s + 16·29-s − 32·37-s + 10·49-s + 16·53-s − 18·81-s + 16·109-s + 16·113-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 8/5·25-s + 2.97·29-s − 5.26·37-s + 10/7·49-s + 2.19·53-s − 2·81-s + 1.53·109-s + 1.50·113-s + 3.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(41923.7\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.085290123\)
\(L(\frac12)\) \(\approx\) \(4.085290123\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
good3$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.57780206148932177669843248804, −6.55745040968756918882947676452, −6.35957068000839944793288951178, −5.81347232657177139814617120271, −5.61375500941123443978476593547, −5.55430642434611354518589245045, −5.39676689209153270887632563842, −5.21434628590146253874550988409, −4.80363124088719428589376227549, −4.62882904224517515245740995066, −4.38150009534384657098008713447, −4.32090201298362643699747114542, −4.12277068882847408310164497820, −3.49681906302410033716260317727, −3.31979739521265682354596216654, −3.28809623886295478360444267694, −3.15454358769357933207399878380, −2.64307360221573083493627350321, −2.51231382936894167438888931944, −2.04039073266894559225728347601, −1.79214795135285516117430169771, −1.67981378046781151188319060897, −1.01050733886054927448279798545, −0.789314099347037547854179069621, −0.44586757487140784037279395617, 0.44586757487140784037279395617, 0.789314099347037547854179069621, 1.01050733886054927448279798545, 1.67981378046781151188319060897, 1.79214795135285516117430169771, 2.04039073266894559225728347601, 2.51231382936894167438888931944, 2.64307360221573083493627350321, 3.15454358769357933207399878380, 3.28809623886295478360444267694, 3.31979739521265682354596216654, 3.49681906302410033716260317727, 4.12277068882847408310164497820, 4.32090201298362643699747114542, 4.38150009534384657098008713447, 4.62882904224517515245740995066, 4.80363124088719428589376227549, 5.21434628590146253874550988409, 5.39676689209153270887632563842, 5.55430642434611354518589245045, 5.61375500941123443978476593547, 5.81347232657177139814617120271, 6.35957068000839944793288951178, 6.55745040968756918882947676452, 6.57780206148932177669843248804

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