Properties

Degree 8
Conductor $ 2^{20} \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  − 8·3-s + 8·9-s − 16·11-s + 72·17-s − 8·19-s + 16·25-s + 152·27-s + 128·33-s − 40·41-s + 80·43-s − 14·49-s − 576·51-s + 64·57-s − 184·59-s + 224·67-s + 232·73-s − 128·75-s − 658·81-s − 88·83-s + 312·89-s − 136·97-s − 128·99-s + 96·107-s − 304·113-s − 180·121-s + 320·123-s + 127-s + ⋯
L(s)  = 1  − 8/3·3-s + 8/9·9-s − 1.45·11-s + 4.23·17-s − 0.421·19-s + 0.639·25-s + 5.62·27-s + 3.87·33-s − 0.975·41-s + 1.86·43-s − 2/7·49-s − 11.2·51-s + 1.12·57-s − 3.11·59-s + 3.34·67-s + 3.17·73-s − 1.70·75-s − 8.12·81-s − 1.06·83-s + 3.50·89-s − 1.40·97-s − 1.29·99-s + 0.897·107-s − 2.69·113-s − 1.48·121-s + 2.60·123-s + 0.00787·127-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{20} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{224} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{20} \cdot 7^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.425459\)
\(L(\frac12)\)  \(\approx\)  \(0.425459\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
good3$D_{4}$ \( ( 1 + 4 T + 20 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
5$D_4\times C_2$ \( 1 - 16 T^{2} - 254 T^{4} - 16 p^{4} T^{6} + p^{8} T^{8} \)
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

−8.569815113634751333028320854399, −8.260404861765974515613790328544, −8.045156681462218737076019202083, −7.993663960780479254245842136764, −7.77614335464481283665981319140, −7.41245061528077777686058289450, −6.85532526023514157462410474096, −6.73846378729020161694477070503, −6.35926626301545781119199235191, −6.12774055583072743141955483198, −5.77676430335292033363517752446, −5.65678107773337871540974453576, −5.38204713638352848394000394657, −5.26355230200992111137123162311, −5.08189643313897293910187252361, −4.74421740273401381882412873870, −4.30917080448361251093015817601, −3.47947332509970968791082260630, −3.36152409000113643669722445606, −3.17456994781437947257308890850, −2.58025481637514242409427015417, −2.27441102717383305559999103030, −1.13927552815464686216722861428, −0.883292174761147313084179093256, −0.32103442146383376525270727863, 0.32103442146383376525270727863, 0.883292174761147313084179093256, 1.13927552815464686216722861428, 2.27441102717383305559999103030, 2.58025481637514242409427015417, 3.17456994781437947257308890850, 3.36152409000113643669722445606, 3.47947332509970968791082260630, 4.30917080448361251093015817601, 4.74421740273401381882412873870, 5.08189643313897293910187252361, 5.26355230200992111137123162311, 5.38204713638352848394000394657, 5.65678107773337871540974453576, 5.77676430335292033363517752446, 6.12774055583072743141955483198, 6.35926626301545781119199235191, 6.73846378729020161694477070503, 6.85532526023514157462410474096, 7.41245061528077777686058289450, 7.77614335464481283665981319140, 7.993663960780479254245842136764, 8.045156681462218737076019202083, 8.260404861765974515613790328544, 8.569815113634751333028320854399

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