# Properties

 Label 8-2160e4-1.1-c2e4-0-5 Degree $8$ Conductor $2.177\times 10^{13}$ Sign $1$ Analytic cond. $1.19992\times 10^{7}$ Root an. cond. $7.67174$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 32·13-s + 10·25-s + 8·37-s + 196·49-s − 124·61-s − 296·73-s − 128·97-s + 388·109-s + 268·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 36·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
 L(s)  = 1 − 2.46·13-s + 2/5·25-s + 8/37·37-s + 4·49-s − 2.03·61-s − 4.05·73-s − 1.31·97-s + 3.55·109-s + 2.21·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.213·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 3^{12} \cdot 5^{4}$$ Sign: $1$ Analytic conductor: $$1.19992\times 10^{7}$$ Root analytic conductor: $$7.67174$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$2.158021922$$ $$L(\frac12)$$ $$\approx$$ $$2.158021922$$ $$L(2)$$ 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 $$1$$
3 $$1$$
5$C_2$ $$( 1 - p T^{2} )^{2}$$
good7$C_1$$\times$$C_1$ $$( 1 - p T )^{4}( 1 + p T )^{4}$$
11$C_2^2$ $$( 1 - 134 T^{2} + p^{4} T^{4} )^{2}$$
13$C_2$ $$( 1 + 8 T + p^{2} T^{2} )^{4}$$
17$C_2^2$ $$( 1 + 533 T^{2} + p^{4} T^{4} )^{2}$$
19$C_2^2$ $$( 1 - 587 T^{2} + p^{4} T^{4} )^{2}$$
23$C_2^2$ $$( 1 - 1031 T^{2} + p^{4} T^{4} )^{2}$$
29$C_2^2$ $$( 1 + 962 T^{2} + p^{4} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - 707 T^{2} + p^{4} T^{4} )^{2}$$
37$C_2$ $$( 1 - 2 T + p^{2} T^{2} )^{4}$$
41$C_2^2$ $$( 1 + 1742 T^{2} + p^{4} T^{4} )^{2}$$
43$C_2^2$ $$( 1 - 3158 T^{2} + p^{4} T^{4} )^{2}$$
47$C_2^2$ $$( 1 - 3986 T^{2} + p^{4} T^{4} )^{2}$$
53$C_2^2$ $$( 1 + 5213 T^{2} + p^{4} T^{4} )^{2}$$
59$C_2^2$ $$( 1 - 5234 T^{2} + p^{4} T^{4} )^{2}$$
61$C_2$ $$( 1 + 31 T + p^{2} T^{2} )^{4}$$
67$C_2^2$ $$( 1 - 4118 T^{2} + p^{4} T^{4} )^{2}$$
71$C_2^2$ $$( 1 - 9974 T^{2} + p^{4} T^{4} )^{2}$$
73$C_2$ $$( 1 + 74 T + p^{2} T^{2} )^{4}$$
79$C_2^2$ $$( 1 - 1547 T^{2} + p^{4} T^{4} )^{2}$$
83$C_2^2$ $$( 1 - 7703 T^{2} + p^{4} T^{4} )^{2}$$
89$C_2^2$ $$( 1 + 11342 T^{2} + p^{4} T^{4} )^{2}$$
97$C_2$ $$( 1 + 32 T + p^{2} T^{2} )^{4}$$
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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

−6.21895743555541327273323448301, −5.94145548463962808516832303388, −5.89927398324106580499741039614, −5.62517647176238080416208205804, −5.51245406969399305721300450409, −5.10895319406599392552444750615, −4.99257406280318301221567393053, −4.75198028039488891325858033454, −4.51717659651994571545481954558, −4.43699503185256766995682961944, −4.18981099596546276331650643148, −3.94377149640341345216257786874, −3.67479878854967632914320046839, −3.29334637581889975181995960688, −3.13028483026825790248267859379, −2.79940763442211789076747369219, −2.70864148126329691178979581925, −2.36514137020532690840035136381, −2.28264846135324403622180586183, −1.77987761084753340742937143830, −1.70625317469290917021147609729, −1.16017625810997476764188873012, −0.937447228610896350048525463048, −0.42948900672542783707939569905, −0.25796796134412198210283112372, 0.25796796134412198210283112372, 0.42948900672542783707939569905, 0.937447228610896350048525463048, 1.16017625810997476764188873012, 1.70625317469290917021147609729, 1.77987761084753340742937143830, 2.28264846135324403622180586183, 2.36514137020532690840035136381, 2.70864148126329691178979581925, 2.79940763442211789076747369219, 3.13028483026825790248267859379, 3.29334637581889975181995960688, 3.67479878854967632914320046839, 3.94377149640341345216257786874, 4.18981099596546276331650643148, 4.43699503185256766995682961944, 4.51717659651994571545481954558, 4.75198028039488891325858033454, 4.99257406280318301221567393053, 5.10895319406599392552444750615, 5.51245406969399305721300450409, 5.62517647176238080416208205804, 5.89927398324106580499741039614, 5.94145548463962808516832303388, 6.21895743555541327273323448301