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Properties

Label 8-600e4-1.1-c0e4-0-1
Degree $8$
Conductor $129600000000$
Sign $1$
Analytic cond. $0.00803958$
Root an. cond. $0.547210$
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-s + 5-s + 6-s − 2·7-s + 10-s − 3·11-s − 2·14-s + 15-s − 2·21-s − 3·22-s + 2·29-s + 30-s + 3·31-s − 32-s − 3·33-s − 2·35-s − 2·42-s + 49-s + 2·53-s − 3·55-s + 2·58-s − 3·59-s + 3·62-s − 64-s − 3·66-s − 2·70-s + ⋯
L(s)  = 1  + 2-s + 3-s + 5-s + 6-s − 2·7-s + 10-s − 3·11-s − 2·14-s + 15-s − 2·21-s − 3·22-s + 2·29-s + 30-s + 3·31-s − 32-s − 3·33-s − 2·35-s − 2·42-s + 49-s + 2·53-s − 3·55-s + 2·58-s − 3·59-s + 3·62-s − 64-s − 3·66-s − 2·70-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(0.00803958\)
Root analytic conductor: \(0.547210\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9983621939\)
\(L(\frac12)\) \(\approx\) \(0.9983621939\)
\(L(1)\) 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$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
3$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
5$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
good7$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
11$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
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_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
29$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
31$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
53$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
59$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
61$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
67$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
71$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
73$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
79$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
89$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
97$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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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

−7.84232125263057793666246361886, −7.66981501816775883330459380997, −7.43258593177356800160634817136, −7.14254561741020520523290292026, −6.97670638188966200650886100836, −6.86649025843642921970021235882, −6.16801099675516020279824416309, −6.02856331054450554983654699819, −6.01834080393432966584567233959, −5.89463443287140223083459360678, −5.69430914124502981247248948137, −5.07288703322210171428009826633, −4.75093527236746108051610999977, −4.73415586335395170227535938226, −4.54041723275158527535589520431, −4.46165948994787499370132581884, −3.67476107723092059868908245222, −3.43394796637693852198834991532, −3.10520709311688864591500395430, −3.03312472714709196783661963924, −2.70509528835814710740052894529, −2.62389902806177915964267520957, −2.15081326373838919591372847816, −1.82616607877136577685655652354, −0.890317674643063229284811383072, 0.890317674643063229284811383072, 1.82616607877136577685655652354, 2.15081326373838919591372847816, 2.62389902806177915964267520957, 2.70509528835814710740052894529, 3.03312472714709196783661963924, 3.10520709311688864591500395430, 3.43394796637693852198834991532, 3.67476107723092059868908245222, 4.46165948994787499370132581884, 4.54041723275158527535589520431, 4.73415586335395170227535938226, 4.75093527236746108051610999977, 5.07288703322210171428009826633, 5.69430914124502981247248948137, 5.89463443287140223083459360678, 6.01834080393432966584567233959, 6.02856331054450554983654699819, 6.16801099675516020279824416309, 6.86649025843642921970021235882, 6.97670638188966200650886100836, 7.14254561741020520523290292026, 7.43258593177356800160634817136, 7.66981501816775883330459380997, 7.84232125263057793666246361886

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