Properties

Label 8-1536e4-1.1-c1e4-0-15
Degree $8$
Conductor $5.566\times 10^{12}$
Sign $1$
Analytic cond. $22629.4$
Root an. cond. $3.50214$
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  − 2·9-s + 8·17-s + 16·25-s − 24·41-s − 24·49-s + 32·73-s + 3·81-s − 8·89-s − 56·97-s + 24·113-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 16·153-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2/3·9-s + 1.94·17-s + 16/5·25-s − 3.74·41-s − 3.42·49-s + 3.74·73-s + 1/3·81-s − 0.847·89-s − 5.68·97-s + 2.25·113-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 − 1.29·153-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{36} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(22629.4\)
Root analytic conductor: \(3.50214\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{36} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.080729561\)
\(L(\frac12)\) \(\approx\) \(3.080729561\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( ( 1 + T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 104 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 140 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
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

−6.76241129180993567421761964775, −6.72864005944816266343006699796, −6.42940377067042922599252292052, −6.03501304535143798109087317742, −5.78573880930752368357860752782, −5.65554863927305572843090020217, −5.42403874782707109560074023926, −5.22487560292815613022424297973, −4.97876488949299956776706366382, −4.73194532904851204774726387330, −4.72164577123277532950718553295, −4.44332357028181874110531765610, −4.01186745391018928641963689062, −3.52474159633929675218895599722, −3.44815816137743851729648616350, −3.41813048048615366335659041423, −2.99992741941636216903736522998, −2.90808180464136359971399772821, −2.69829836843515740546085507043, −2.05628216994057905664106507850, −1.90858284224124291061078531062, −1.41319520582790997952087696415, −1.36530335437389536355285169439, −0.74586506034854413036551954923, −0.40478530864035719151949998930, 0.40478530864035719151949998930, 0.74586506034854413036551954923, 1.36530335437389536355285169439, 1.41319520582790997952087696415, 1.90858284224124291061078531062, 2.05628216994057905664106507850, 2.69829836843515740546085507043, 2.90808180464136359971399772821, 2.99992741941636216903736522998, 3.41813048048615366335659041423, 3.44815816137743851729648616350, 3.52474159633929675218895599722, 4.01186745391018928641963689062, 4.44332357028181874110531765610, 4.72164577123277532950718553295, 4.73194532904851204774726387330, 4.97876488949299956776706366382, 5.22487560292815613022424297973, 5.42403874782707109560074023926, 5.65554863927305572843090020217, 5.78573880930752368357860752782, 6.03501304535143798109087317742, 6.42940377067042922599252292052, 6.72864005944816266343006699796, 6.76241129180993567421761964775

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