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Properties

Label 8-1045e4-1.1-c0e4-0-1
Degree $8$
Conductor $1.193\times 10^{12}$
Sign $1$
Analytic cond. $0.0739764$
Root an. cond. $0.722165$
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  + 4-s − 5-s − 5·7-s + 9-s + 11-s + 5·17-s − 19-s − 20-s − 5·28-s + 5·35-s + 36-s + 44-s − 45-s + 14·49-s − 55-s − 3·61-s − 5·63-s + 5·68-s − 76-s − 5·77-s − 5·85-s + 95-s + 99-s + 2·101-s − 25·119-s + 127-s + 131-s + ⋯
L(s)  = 1  + 4-s − 5-s − 5·7-s + 9-s + 11-s + 5·17-s − 19-s − 20-s − 5·28-s + 5·35-s + 36-s + 44-s − 45-s + 14·49-s − 55-s − 3·61-s − 5·63-s + 5·68-s − 76-s − 5·77-s − 5·85-s + 95-s + 99-s + 2·101-s − 25·119-s + 127-s + 131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 11^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(0.0739764\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{4} \cdot 11^{4} \cdot 19^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6179261772\)
\(L(\frac12)\) \(\approx\) \(0.6179261772\)
\(L(1)\) 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)$
bad5$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
11$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
19$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
good2$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
3$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
7$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
17$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
23$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
29$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
31$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
41$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
43$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
47$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
53$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
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$C_2$ \( ( 1 + T^{2} )^{4} \)
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$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
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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

−7.30210656246040789630173681309, −7.06375358034902341884526212948, −6.89080000509376905665192877996, −6.61250716204408212517590661979, −6.41811534395533462293071167400, −6.32221189453030459829490628738, −6.01384794047537770724602692529, −5.88618136283352705684589311026, −5.76535839848531155309879708630, −5.38092857440446642405383974307, −5.28984435768582706539144319850, −4.56323279759636789708873396419, −4.48715950418275564933690547249, −4.07639387084559780013526063422, −3.70414988938447281004788445922, −3.65237288046653406491478526048, −3.63249681725534015376406021571, −3.19237767101472123263396280622, −2.93456754162108800685458813777, −2.90900610716713555444577373122, −2.79806462571601789785699824968, −1.86155229764754625425459855511, −1.79350393639726301672892269740, −1.01794471533447493018936847108, −0.72978542713341857223768524333, 0.72978542713341857223768524333, 1.01794471533447493018936847108, 1.79350393639726301672892269740, 1.86155229764754625425459855511, 2.79806462571601789785699824968, 2.90900610716713555444577373122, 2.93456754162108800685458813777, 3.19237767101472123263396280622, 3.63249681725534015376406021571, 3.65237288046653406491478526048, 3.70414988938447281004788445922, 4.07639387084559780013526063422, 4.48715950418275564933690547249, 4.56323279759636789708873396419, 5.28984435768582706539144319850, 5.38092857440446642405383974307, 5.76535839848531155309879708630, 5.88618136283352705684589311026, 6.01384794047537770724602692529, 6.32221189453030459829490628738, 6.41811534395533462293071167400, 6.61250716204408212517590661979, 6.89080000509376905665192877996, 7.06375358034902341884526212948, 7.30210656246040789630173681309

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