Properties

Label 8-1792e4-1.1-c1e4-0-0
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·9-s + 20·25-s − 14·49-s + 90·81-s − 8·113-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 240·225-s + 227-s + ⋯
L(s)  = 1  − 4·9-s + 4·25-s − 2·49-s + 10·81-s − 0.752·113-s + 1.09·121-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 + 4·169-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 + 0.0669·223-s − 16·225-s + 0.0663·227-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

\(L(1)\) \(\approx\) \(0.3820019647\)
\(L(\frac12)\) \(\approx\) \(0.3820019647\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{4} \)
5$C_2$ \( ( 1 - p T^{2} )^{4} \)
11$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
29$C_2^2$ \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2$ \( ( 1 - p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - p T^{2} )^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2$ \( ( 1 - p T^{2} )^{4} \)
97$C_2$ \( ( 1 - p T^{2} )^{4} \)
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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

−6.51124727178665106464595759833, −6.43126091711712095165456060778, −6.14990461490122909460150876560, −6.10224948132327949156146112607, −5.68170596021856925302131670135, −5.49484127351967012490516364774, −5.33402158100224379760996070251, −5.09124561002572294023542633619, −5.00962075529077404242404454878, −4.84748591031621635975139464903, −4.40840005774288851798571089458, −4.37596182529510836722824944579, −3.78242066559519555049596916387, −3.58330950402732531144692441401, −3.35340519501085861507139776405, −3.12538750554399393355334830493, −2.95001228245309081614793449953, −2.68253964550252967876753846398, −2.66059392762584145875861190976, −2.31157841765967597280528847082, −1.89310544637193959402721737601, −1.52876288927692105891439175842, −0.890599274575855586973926834529, −0.830916243853884145185439778155, −0.13921922143707051499433477077, 0.13921922143707051499433477077, 0.830916243853884145185439778155, 0.890599274575855586973926834529, 1.52876288927692105891439175842, 1.89310544637193959402721737601, 2.31157841765967597280528847082, 2.66059392762584145875861190976, 2.68253964550252967876753846398, 2.95001228245309081614793449953, 3.12538750554399393355334830493, 3.35340519501085861507139776405, 3.58330950402732531144692441401, 3.78242066559519555049596916387, 4.37596182529510836722824944579, 4.40840005774288851798571089458, 4.84748591031621635975139464903, 5.00962075529077404242404454878, 5.09124561002572294023542633619, 5.33402158100224379760996070251, 5.49484127351967012490516364774, 5.68170596021856925302131670135, 6.10224948132327949156146112607, 6.14990461490122909460150876560, 6.43126091711712095165456060778, 6.51124727178665106464595759833

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