Properties

Label 8-384e4-1.1-c2e4-0-2
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $11985.7$
Root an. cond. $3.23469$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 6·9-s − 72·17-s + 68·25-s + 72·41-s + 100·49-s − 328·73-s + 27·81-s + 504·89-s + 440·97-s − 504·113-s − 388·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 432·153-s + 157-s + 163-s + 167-s + 676·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2/3·9-s − 4.23·17-s + 2.71·25-s + 1.75·41-s + 2.04·49-s − 4.49·73-s + 1/3·81-s + 5.66·89-s + 4.53·97-s − 4.46·113-s − 3.20·121-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 − 2.82·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 4·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(11985.7\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.877496290\)
\(L(\frac12)\) \(\approx\) \(2.877496290\)
\(L(2)\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - 34 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 50 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 194 T^{2} + p^{4} T^{4} )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
17$C_2$ \( ( 1 + 18 T + p^{2} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + 290 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 670 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 1666 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 430 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 2446 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 18 T + p^{2} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 190 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 2690 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 3682 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 3074 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 2258 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 8546 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 8354 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 82 T + p^{2} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 8594 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 3550 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 126 T + p^{2} T^{2} )^{4} \)
97$C_2$ \( ( 1 - 110 T + p^{2} T^{2} )^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.83467195774336091034254854193, −7.66498121682739745034298865301, −7.56553605493569526247158035262, −7.12833485260679962922957985679, −7.04548469267014271353779973728, −6.68087479353083523481473786644, −6.45622806814619324450059124112, −6.27392022834307555599641346769, −6.25609057044065710435807964904, −5.68459333959429084492062827074, −5.33239788539587570502146866863, −4.96186869166213398999507704602, −4.84188599838223589715607340585, −4.55049271045132023628077433739, −4.22003201048297053337997931721, −4.09998419567446782834805318384, −3.89001077513743863548863622670, −3.10915964936445703698215728781, −3.02454273033745894996260135960, −2.45595250211152728058045069491, −2.36685497314927407665575740552, −1.95232507248799730840320909591, −1.46485556768182803787052876897, −0.78042646156902315649360551355, −0.44727877567296724055635898644, 0.44727877567296724055635898644, 0.78042646156902315649360551355, 1.46485556768182803787052876897, 1.95232507248799730840320909591, 2.36685497314927407665575740552, 2.45595250211152728058045069491, 3.02454273033745894996260135960, 3.10915964936445703698215728781, 3.89001077513743863548863622670, 4.09998419567446782834805318384, 4.22003201048297053337997931721, 4.55049271045132023628077433739, 4.84188599838223589715607340585, 4.96186869166213398999507704602, 5.33239788539587570502146866863, 5.68459333959429084492062827074, 6.25609057044065710435807964904, 6.27392022834307555599641346769, 6.45622806814619324450059124112, 6.68087479353083523481473786644, 7.04548469267014271353779973728, 7.12833485260679962922957985679, 7.56553605493569526247158035262, 7.66498121682739745034298865301, 7.83467195774336091034254854193

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