Properties

Degree 8
Conductor $ 2^{20} \cdot 3^{12} \cdot 7^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 4·7-s + 8·13-s − 2·17-s + 8·19-s − 25-s − 6·29-s + 6·31-s − 8·35-s + 6·37-s + 6·41-s − 2·43-s + 2·47-s + 10·49-s − 6·53-s + 4·61-s − 16·65-s − 4·67-s + 16·71-s + 12·73-s − 2·79-s − 18·83-s + 4·85-s + 8·89-s + 32·91-s − 16·95-s + 24·97-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.51·7-s + 2.21·13-s − 0.485·17-s + 1.83·19-s − 1/5·25-s − 1.11·29-s + 1.07·31-s − 1.35·35-s + 0.986·37-s + 0.937·41-s − 0.304·43-s + 0.291·47-s + 10/7·49-s − 0.824·53-s + 0.512·61-s − 1.98·65-s − 0.488·67-s + 1.89·71-s + 1.40·73-s − 0.225·79-s − 1.97·83-s + 0.433·85-s + 0.847·89-s + 3.35·91-s − 1.64·95-s + 2.43·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{20} \cdot 3^{12} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{20} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $10.01809984$
$L(\frac12)$  $\approx$  $10.01809984$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7$C_1$ \( ( 1 - T )^{4} \)
good5$C_2 \wr S_4$ \( 1 + 2 T + p T^{2} + 18 T^{3} + 48 T^{4} + 18 p T^{5} + p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 8 T^{2} - 16 T^{3} + 126 T^{4} - 16 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 8 T + 31 T^{2} - 128 T^{3} + 556 T^{4} - 128 p T^{5} + 31 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 2 T + 20 T^{2} - 52 T^{3} + 13 T^{4} - 52 p T^{5} + 20 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 8 T + 64 T^{2} - 360 T^{3} + 1806 T^{4} - 360 p T^{5} + 64 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 47 T^{2} + 128 T^{3} + 1008 T^{4} + 128 p T^{5} + 47 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 6 T + 113 T^{2} + 490 T^{3} + 4896 T^{4} + 490 p T^{5} + 113 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 6 T + 121 T^{2} - 526 T^{3} + 5604 T^{4} - 526 p T^{5} + 121 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 6 T + 97 T^{2} - 262 T^{3} + 3804 T^{4} - 262 p T^{5} + 97 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 6 T + 161 T^{2} - 698 T^{3} + 9852 T^{4} - 698 p T^{5} + 161 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 2 T + 100 T^{2} - 104 T^{3} + 4469 T^{4} - 104 p T^{5} + 100 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 2 T + 41 T^{2} - 146 T^{3} + 3640 T^{4} - 146 p T^{5} + 41 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 6 T + p T^{2} - 370 T^{3} - 2412 T^{4} - 370 p T^{5} + p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 170 T^{2} - 32 T^{3} + 13899 T^{4} - 32 p T^{5} + 170 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 4 T + 124 T^{2} + 116 T^{3} + 6358 T^{4} + 116 p T^{5} + 124 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 4 T + 139 T^{2} + 892 T^{3} + 10912 T^{4} + 892 p T^{5} + 139 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 16 T + 335 T^{2} - 3432 T^{3} + 37440 T^{4} - 3432 p T^{5} + 335 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 12 T + 184 T^{2} - 2148 T^{3} + 16782 T^{4} - 2148 p T^{5} + 184 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
79$S_4\times C_2$ \( 1 + 2 T + 97 T^{2} - 1498 T^{3} - 1952 T^{4} - 1498 p T^{5} + 97 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 18 T + 389 T^{2} + 4490 T^{3} + 50748 T^{4} + 4490 p T^{5} + 389 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 8 T + 263 T^{2} - 1500 T^{3} + 30168 T^{4} - 1500 p T^{5} + 263 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 24 T + 460 T^{2} - 5704 T^{3} + 65814 T^{4} - 5704 p T^{5} + 460 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−5.84929879268968449479614102728, −5.40412128366557831389440500139, −5.33113205950624454297254971020, −5.01407837492605167896079487136, −4.92626429629227693811661022731, −4.66145346698207603098709566360, −4.62058999487592857434156804450, −4.28077577935021345627136185319, −4.25046878206362203332455427209, −3.85712473997849002561051989550, −3.64135724663257779763802136880, −3.60916245986279462617032984587, −3.46896820614003616419620174203, −3.27177906219288173617550890745, −2.82119765943172021065744862644, −2.75724878642591950797386413873, −2.46355349101495931165314644410, −2.04253846886696217174063333670, −2.01332695580953978923732368287, −1.65759641592799267567329070426, −1.52590446060634445460977134049, −1.08163618158566882270727403341, −0.810036351205887702786946299486, −0.790265105800648355745738227065, −0.41303438837497041173464778430, 0.41303438837497041173464778430, 0.790265105800648355745738227065, 0.810036351205887702786946299486, 1.08163618158566882270727403341, 1.52590446060634445460977134049, 1.65759641592799267567329070426, 2.01332695580953978923732368287, 2.04253846886696217174063333670, 2.46355349101495931165314644410, 2.75724878642591950797386413873, 2.82119765943172021065744862644, 3.27177906219288173617550890745, 3.46896820614003616419620174203, 3.60916245986279462617032984587, 3.64135724663257779763802136880, 3.85712473997849002561051989550, 4.25046878206362203332455427209, 4.28077577935021345627136185319, 4.62058999487592857434156804450, 4.66145346698207603098709566360, 4.92626429629227693811661022731, 5.01407837492605167896079487136, 5.33113205950624454297254971020, 5.40412128366557831389440500139, 5.84929879268968449479614102728

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