Properties

Label 8-63e8-1.1-c0e4-0-1
Degree $8$
Conductor $2.482\times 10^{14}$
Sign $1$
Analytic cond. $15.3940$
Root an. cond. $1.40740$
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  − 2·16-s − 2·25-s − 8·67-s + 8·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 2·16-s − 2·25-s − 8·67-s + 8·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(15.3940\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3969} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{16} \cdot 7^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3744571302\)
\(L(\frac12)\) \(\approx\) \(0.3744571302\)
\(L(1)\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( ( 1 + T^{4} )^{2} \)
5$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
11$C_2^3$ \( 1 - T^{4} + T^{8} \)
13$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
19$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
23$C_2^3$ \( 1 - T^{4} + T^{8} \)
29$C_2^3$ \( 1 - T^{4} + T^{8} \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
41$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
43$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
53$C_2^3$ \( 1 - T^{4} + T^{8} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{4} \)
67$C_1$ \( ( 1 + T )^{8} \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
79$C_1$ \( ( 1 - T )^{8} \)
83$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
97$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
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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

−6.12281662082391882242330802817, −6.06602691552107228512197709700, −5.81624480162639099685164738764, −5.61571589940076969099699448195, −5.35983666612812245471815012777, −5.05252172679609001610035991608, −5.02442634269376999756962356618, −4.78594537069317133141177355301, −4.57802350550996232162403728751, −4.34686779412842207822431867537, −4.19051879121676900180348697477, −3.97426676866733277205237271218, −3.71052475631998789304133246631, −3.64657663459570219428192199186, −3.28656304363392551568212806775, −2.99499326994217563196980736520, −2.78622285465543087952491659906, −2.64447363936065576326518963279, −2.34734539959616413010260208756, −1.88940739241199898980360440529, −1.86013173053837737019475651569, −1.83859264676624976910908626637, −1.19365289892052036394689059024, −0.989378628559122781933709126733, −0.21760256341572365171808533235, 0.21760256341572365171808533235, 0.989378628559122781933709126733, 1.19365289892052036394689059024, 1.83859264676624976910908626637, 1.86013173053837737019475651569, 1.88940739241199898980360440529, 2.34734539959616413010260208756, 2.64447363936065576326518963279, 2.78622285465543087952491659906, 2.99499326994217563196980736520, 3.28656304363392551568212806775, 3.64657663459570219428192199186, 3.71052475631998789304133246631, 3.97426676866733277205237271218, 4.19051879121676900180348697477, 4.34686779412842207822431867537, 4.57802350550996232162403728751, 4.78594537069317133141177355301, 5.02442634269376999756962356618, 5.05252172679609001610035991608, 5.35983666612812245471815012777, 5.61571589940076969099699448195, 5.81624480162639099685164738764, 6.06602691552107228512197709700, 6.12281662082391882242330802817

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