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Properties

Label 8-50e8-1.1-c0e4-0-4
Degree $8$
Conductor $3.906\times 10^{13}$
Sign $1$
Analytic cond. $2.42319$
Root an. cond. $1.11698$
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-s + 3·3-s − 3·6-s − 2·7-s + 6·9-s + 2·14-s − 6·18-s − 6·21-s + 3·23-s + 10·27-s + 3·29-s + 32-s − 2·41-s + 6·42-s − 2·43-s − 3·46-s + 3·47-s + 49-s − 10·54-s − 3·58-s + 3·61-s − 12·63-s − 64-s − 2·67-s + 9·69-s + 15·81-s + 2·82-s + ⋯
L(s)  = 1  − 2-s + 3·3-s − 3·6-s − 2·7-s + 6·9-s + 2·14-s − 6·18-s − 6·21-s + 3·23-s + 10·27-s + 3·29-s + 32-s − 2·41-s + 6·42-s − 2·43-s − 3·46-s + 3·47-s + 49-s − 10·54-s − 3·58-s + 3·61-s − 12·63-s − 64-s − 2·67-s + 9·69-s + 15·81-s + 2·82-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(2.42319\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{16} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.362461516\)
\(L(\frac12)\) \(\approx\) \(2.362461516\)
\(L(1)\) 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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
5 \( 1 \)
good3$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
7$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
11$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
17$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
19$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
23$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
29$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
31$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
41$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
47$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
53$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
59$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
61$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
67$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
71$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
73$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
79$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
89$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
97$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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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

−6.80957694720395977291560723387, −6.42569545697976257439607982438, −6.38795846868676664237777308976, −6.32045039261643648421527118447, −5.50609553873836388668406443944, −5.42684274384342620847585226531, −5.28137136850796864545374456611, −5.12503931356421975676221739015, −4.72131924279114865200428491182, −4.40351728567085114052135257959, −4.35078400151346468009951839520, −4.15743455975078875155715483799, −3.93088397965880431387320748440, −3.41831115214839061594637993902, −3.39846691240787527656508361330, −3.24323451761269354755063409380, −3.03629575700486161031951516084, −2.64698304962521730189400228383, −2.60656313457059585230521389098, −2.60471948355972654842936781932, −2.15275931680747985772103235157, −1.55779364290027309544582065148, −1.37174479817972640737164382247, −1.10339695037839386158293914189, −0.75149585213663157042123248289, 0.75149585213663157042123248289, 1.10339695037839386158293914189, 1.37174479817972640737164382247, 1.55779364290027309544582065148, 2.15275931680747985772103235157, 2.60471948355972654842936781932, 2.60656313457059585230521389098, 2.64698304962521730189400228383, 3.03629575700486161031951516084, 3.24323451761269354755063409380, 3.39846691240787527656508361330, 3.41831115214839061594637993902, 3.93088397965880431387320748440, 4.15743455975078875155715483799, 4.35078400151346468009951839520, 4.40351728567085114052135257959, 4.72131924279114865200428491182, 5.12503931356421975676221739015, 5.28137136850796864545374456611, 5.42684274384342620847585226531, 5.50609553873836388668406443944, 6.32045039261643648421527118447, 6.38795846868676664237777308976, 6.42569545697976257439607982438, 6.80957694720395977291560723387

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