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Properties

Label 8-3332e4-1.1-c0e4-0-1
Degree $8$
Conductor $1.233\times 10^{14}$
Sign $1$
Analytic cond. $7.64624$
Root an. cond. $1.28952$
Motivic weight $0$
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  − 16-s − 2·25-s + 4·37-s − 4·41-s − 4·53-s − 4·61-s − 8·101-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 16-s − 2·25-s + 4·37-s − 4·41-s − 4·53-s − 4·61-s − 8·101-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 7^{8} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(7.64624\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 7^{8} \cdot 17^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02504905152\)
\(L(\frac12)\) \(\approx\) \(0.02504905152\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
7 \( 1 \)
17$C_2^2$ \( 1 + T^{4} \)
good3$C_4\times C_2$ \( 1 + T^{8} \)
5$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
11$C_4\times C_2$ \( 1 + T^{8} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_4\times C_2$ \( 1 + T^{8} \)
29$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
31$C_4\times C_2$ \( 1 + T^{8} \)
37$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
41$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
71$C_4\times C_2$ \( 1 + T^{8} \)
73$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
79$C_4\times C_2$ \( 1 + T^{8} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
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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

−6.37254457185783227428583283178, −6.22901000901665781839665755070, −5.85304469049485541004284379128, −5.63198612189880790730157045095, −5.61288459825055659886386959687, −5.26899848961023729087418598942, −5.16696007834147954951335190888, −4.75249500590844874976955506936, −4.67807656195524327384487625290, −4.42526781571625400715926551456, −4.35716776850597121711549830772, −4.05975409333704124869233484045, −3.89605288478848855727840657011, −3.62707696532172022020501322927, −3.41153363763487267252783706771, −2.90474165750137057727457606987, −2.83461457848544409023271900029, −2.75360915753655870782156684350, −2.71118388306182570542154652944, −1.94827965416770219397599279817, −1.75546052181130333540948532299, −1.73998008984554966297433211330, −1.37071762089987271138999012204, −1.11342710516643518288797401479, −0.05679734376447411383845962127, 0.05679734376447411383845962127, 1.11342710516643518288797401479, 1.37071762089987271138999012204, 1.73998008984554966297433211330, 1.75546052181130333540948532299, 1.94827965416770219397599279817, 2.71118388306182570542154652944, 2.75360915753655870782156684350, 2.83461457848544409023271900029, 2.90474165750137057727457606987, 3.41153363763487267252783706771, 3.62707696532172022020501322927, 3.89605288478848855727840657011, 4.05975409333704124869233484045, 4.35716776850597121711549830772, 4.42526781571625400715926551456, 4.67807656195524327384487625290, 4.75249500590844874976955506936, 5.16696007834147954951335190888, 5.26899848961023729087418598942, 5.61288459825055659886386959687, 5.63198612189880790730157045095, 5.85304469049485541004284379128, 6.22901000901665781839665755070, 6.37254457185783227428583283178

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