Properties

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

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 16·11-s − 72·17-s − 8·19-s + 16·25-s + 40·41-s + 80·43-s − 14·49-s + 184·59-s + 224·67-s + 232·73-s + 88·83-s − 312·89-s − 136·97-s − 96·107-s + 304·113-s − 180·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 592·169-s + 173-s + ⋯
L(s)  = 1  + 1.45·11-s − 4.23·17-s − 0.421·19-s + 0.639·25-s + 0.975·41-s + 1.86·43-s − 2/7·49-s + 3.11·59-s + 3.34·67-s + 3.17·73-s + 1.06·83-s − 3.50·89-s − 1.40·97-s − 0.897·107-s + 2.69·113-s − 1.48·121-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 + 3.50·169-s + 0.00578·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{20} \cdot 3^{8} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{2016} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{20} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(4.100479064\)
\(L(\frac12)\)  \(\approx\)  \(4.100479064\)
\(L(2)\)   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$C_2$ \( ( 1 + p T^{2} )^{2} \)
good5$D_4\times C_2$ \( 1 - 16 T^{2} - 254 T^{4} - 16 p^{4} T^{6} + p^{8} T^{8} \)
11$D_{4}$ \( ( 1 - 8 T + 186 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 592 T^{2} + 143170 T^{4} - 592 p^{4} T^{6} + p^{8} T^{8} \)
17$D_{4}$ \( ( 1 + 36 T + 870 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 + 4 T + 4 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 268 T^{2} + 477286 T^{4} - 268 p^{4} T^{6} + p^{8} T^{8} \)
29$C_2^2$ \( ( 1 - 1178 T^{2} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 1380 T^{2} + 1419974 T^{4} - 1380 p^{4} T^{6} + p^{8} T^{8} \)
37$D_4\times C_2$ \( 1 - 1780 T^{2} + 2031622 T^{4} - 1780 p^{4} T^{6} + p^{8} T^{8} \)
41$D_{4}$ \( ( 1 - 20 T + 3174 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 40 T + 4090 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 7492 T^{2} + 23390470 T^{4} - 7492 p^{4} T^{6} + p^{8} T^{8} \)
53$D_4\times C_2$ \( 1 - 1604 T^{2} - 6155034 T^{4} - 1604 p^{4} T^{6} + p^{8} T^{8} \)
59$D_{4}$ \( ( 1 - 92 T + 8836 T^{2} - 92 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - 13232 T^{2} + 71110338 T^{4} - 13232 p^{4} T^{6} + p^{8} T^{8} \)
67$D_{4}$ \( ( 1 - 112 T + 11602 T^{2} - 112 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 9412 T^{2} + 47279686 T^{4} - 9412 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 116 T + 13894 T^{2} - 116 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 16900 T^{2} + 142880134 T^{4} - 16900 p^{4} T^{6} + p^{8} T^{8} \)
83$D_{4}$ \( ( 1 - 44 T + 13924 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 + 156 T + 20774 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 68 T + 3046 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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

−6.57065785766242096724336441415, −6.16351660299035230804707268157, −5.96921217836367218054909379179, −5.66896901177805412886437250745, −5.47523711090579121812834759080, −5.16023828800970812742764052520, −5.06637021885440238671822895099, −4.88833389682195114279930006533, −4.38585497325716329249657817517, −4.24926564155713104994021166025, −4.21695805374762711099339458163, −3.93546265980307474265407173445, −3.90137888536371055315970239886, −3.64162190836668500940307010249, −3.09288602277313798291250248819, −2.89554814856293800843005706157, −2.55967722697070426605631371167, −2.31013698089092948857975947442, −2.26741916803716910060841563111, −1.86761538093128545670378044296, −1.79278876231269490385725203916, −1.18077856323363786701161796721, −0.75524526519699635039338133128, −0.69154894167603822063880637661, −0.28273704292989391192593916218, 0.28273704292989391192593916218, 0.69154894167603822063880637661, 0.75524526519699635039338133128, 1.18077856323363786701161796721, 1.79278876231269490385725203916, 1.86761538093128545670378044296, 2.26741916803716910060841563111, 2.31013698089092948857975947442, 2.55967722697070426605631371167, 2.89554814856293800843005706157, 3.09288602277313798291250248819, 3.64162190836668500940307010249, 3.90137888536371055315970239886, 3.93546265980307474265407173445, 4.21695805374762711099339458163, 4.24926564155713104994021166025, 4.38585497325716329249657817517, 4.88833389682195114279930006533, 5.06637021885440238671822895099, 5.16023828800970812742764052520, 5.47523711090579121812834759080, 5.66896901177805412886437250745, 5.96921217836367218054909379179, 6.16351660299035230804707268157, 6.57065785766242096724336441415

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