Properties

Label 8-1216e4-1.1-c1e4-0-11
Degree $8$
Conductor $2.186\times 10^{12}$
Sign $1$
Analytic cond. $8888.79$
Root an. cond. $3.11605$
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  + 10·9-s − 12·17-s − 4·25-s − 22·49-s − 4·73-s + 57·81-s + 72·89-s − 56·97-s + 24·113-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 120·153-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 10/3·9-s − 2.91·17-s − 4/5·25-s − 3.14·49-s − 0.468·73-s + 19/3·81-s + 7.63·89-s − 5.68·97-s + 2.25·113-s + 4·121-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 − 9.70·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.153·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(8888.79\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.107963332\)
\(L(\frac12)\) \(\approx\) \(4.107963332\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 79 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{4} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{4} \)
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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

−6.84619942592678888942919242058, −6.83927656353014488929483641243, −6.77831619077831555211479370128, −6.22929062557113963510343636474, −6.04354071914397119737973510857, −6.01880978994877518340702791784, −5.90208490320469562409586990325, −5.08185145654037014623175533490, −5.04860448240018613682534738912, −4.81557542890447627150786534131, −4.62019438664337834838522489439, −4.56267758728510822804505539546, −4.29510467888817761876833666905, −3.92553215443835919497526020369, −3.78146532603560973958805244249, −3.63761215572347329031137936426, −3.10761673108875031430359833063, −3.01868337391652168154813455937, −2.45550167055819547236929695384, −1.99570936840885491694436235401, −1.99038635616161493495047140663, −1.66640161108249803549929747609, −1.57573625160565060131243997498, −0.73052321173957581962420246312, −0.52327539071794820116424229984, 0.52327539071794820116424229984, 0.73052321173957581962420246312, 1.57573625160565060131243997498, 1.66640161108249803549929747609, 1.99038635616161493495047140663, 1.99570936840885491694436235401, 2.45550167055819547236929695384, 3.01868337391652168154813455937, 3.10761673108875031430359833063, 3.63761215572347329031137936426, 3.78146532603560973958805244249, 3.92553215443835919497526020369, 4.29510467888817761876833666905, 4.56267758728510822804505539546, 4.62019438664337834838522489439, 4.81557542890447627150786534131, 5.04860448240018613682534738912, 5.08185145654037014623175533490, 5.90208490320469562409586990325, 6.01880978994877518340702791784, 6.04354071914397119737973510857, 6.22929062557113963510343636474, 6.77831619077831555211479370128, 6.83927656353014488929483641243, 6.84619942592678888942919242058

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