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Properties

Label 8-2016e4-1.1-c0e4-0-1
Degree $8$
Conductor $1.652\times 10^{13}$
Sign $1$
Analytic cond. $1.02468$
Root an. cond. $1.00305$
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  − 2·4-s − 4·11-s + 3·16-s − 4·29-s + 4·37-s + 8·44-s − 4·53-s − 4·64-s − 4·67-s + 4·71-s + 4·107-s + 8·116-s + 10·121-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 12·176-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2·4-s − 4·11-s + 3·16-s − 4·29-s + 4·37-s + 8·44-s − 4·53-s − 4·64-s − 4·67-s + 4·71-s + 4·107-s + 8·116-s + 10·121-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 12·176-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.02468\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2292026417\)
\(L(\frac12)\) \(\approx\) \(0.2292026417\)
\(L(1)\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3 \( 1 \)
7$C_2^2$ \( 1 + T^{4} \)
good5$C_4\times C_2$ \( 1 + T^{8} \)
11$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
13$C_4\times C_2$ \( 1 + T^{8} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
19$C_4\times C_2$ \( 1 + T^{8} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_1$$\times$$C_2^2$ \( ( 1 - T )^{4}( 1 + T^{4} ) \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
59$C_4\times C_2$ \( 1 + T^{8} \)
61$C_4\times C_2$ \( 1 + T^{8} \)
67$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
71$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + T^{4} )^{2} \)
83$C_4\times C_2$ \( 1 + T^{8} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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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

−6.54137597134582445942489343045, −6.31660536756431502932606478048, −6.18259164098407931160302873731, −5.89207665953990287976433038248, −5.81026686113861551141785463727, −5.72902113038716905034427080292, −5.26900824355964472708223216374, −5.22029618327676392781581150953, −5.02456842838157491534740933452, −4.80327268371171929098204209343, −4.64232388546465081000272196256, −4.49475995485367125984307323852, −4.17854997823022952060574880953, −3.82672714454300846548913774749, −3.75981324920675879534514798372, −3.31467915592426290155433760809, −3.16485132385389032180287234293, −2.99054064300902038848211216543, −2.60494098744945403298028760799, −2.52728407753931018871541554280, −1.93035341630232667844571464797, −1.90083434302803730574946906646, −1.45534177058383156583951114427, −0.75272173311407154287499577779, −0.34391161518839371166495407209, 0.34391161518839371166495407209, 0.75272173311407154287499577779, 1.45534177058383156583951114427, 1.90083434302803730574946906646, 1.93035341630232667844571464797, 2.52728407753931018871541554280, 2.60494098744945403298028760799, 2.99054064300902038848211216543, 3.16485132385389032180287234293, 3.31467915592426290155433760809, 3.75981324920675879534514798372, 3.82672714454300846548913774749, 4.17854997823022952060574880953, 4.49475995485367125984307323852, 4.64232388546465081000272196256, 4.80327268371171929098204209343, 5.02456842838157491534740933452, 5.22029618327676392781581150953, 5.26900824355964472708223216374, 5.72902113038716905034427080292, 5.81026686113861551141785463727, 5.89207665953990287976433038248, 6.18259164098407931160302873731, 6.31660536756431502932606478048, 6.54137597134582445942489343045

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