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

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Normalization:  

Dirichlet series

L(s)  = 1  − 18·5-s + 7·9-s + 64·13-s + 14·17-s + 131·25-s − 128·29-s + 2·37-s − 160·41-s − 126·45-s + 94·49-s − 14·53-s − 158·61-s − 1.15e3·65-s − 286·73-s + 81·81-s − 252·85-s + 194·89-s − 352·97-s + 30·101-s − 226·109-s + 224·113-s + 448·117-s − 233·121-s − 342·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 3.59·5-s + 7/9·9-s + 4.92·13-s + 0.823·17-s + 5.23·25-s − 4.41·29-s + 2/37·37-s − 3.90·41-s − 2.79·45-s + 1.91·49-s − 0.264·53-s − 2.59·61-s − 17.7·65-s − 3.91·73-s + 81-s − 2.96·85-s + 2.17·89-s − 3.62·97-s + 0.297·101-s − 2.07·109-s + 1.98·113-s + 3.82·117-s − 1.92·121-s − 2.73·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-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  =  \((8,\ 2^{20} \cdot 7^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0469269\)
\(L(\frac12)\)  \(\approx\)  \(0.0469269\)
\(L(2)\)   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$C_2^2$ \( 1 - 94 T^{2} + p^{4} T^{4} \)
good3$C_2^3$ \( 1 - 7 T^{2} - 32 T^{4} - 7 p^{4} T^{6} + p^{8} T^{8} \)
5$C_2^2$ \( ( 1 + 9 T + 56 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
11$C_2^3$ \( 1 + 233 T^{2} + 39648 T^{4} + 233 p^{4} T^{6} + p^{8} T^{8} \)
13$C_2$ \( ( 1 - 16 T + p^{2} T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 7 T - 240 T^{2} - 7 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$C_2^2$$\times$$C_2^2$ \( ( 1 - 46 T^{2} + p^{4} T^{4} )( 1 + 647 T^{2} + p^{4} T^{4} ) \)
23$C_2^3$ \( 1 + 697 T^{2} + 205968 T^{4} + 697 p^{4} T^{6} + p^{8} T^{8} \)
29$C_2$ \( ( 1 + 32 T + p^{2} T^{2} )^{4} \)
31$C_2^3$ \( 1 + 1801 T^{2} + 2320080 T^{4} + 1801 p^{4} T^{6} + p^{8} T^{8} \)
37$C_2^2$ \( ( 1 - T - 1368 T^{2} - p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 40 T + p^{2} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 2098 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 2807 T^{2} + 2999568 T^{4} - 2807 p^{4} T^{6} + p^{8} T^{8} \)
53$C_2^2$ \( ( 1 + 7 T - 2760 T^{2} + 7 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$C_2^3$ \( 1 + 4153 T^{2} + 5130048 T^{4} + 4153 p^{4} T^{6} + p^{8} T^{8} \)
61$C_2^2$ \( ( 1 + 79 T + 2520 T^{2} + 79 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 8857 T^{2} + 58295328 T^{4} + 8857 p^{4} T^{6} + p^{8} T^{8} \)
71$C_2^2$ \( ( 1 - 7778 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 46 T + p^{2} T^{2} )^{2}( 1 + 97 T + p^{2} T^{2} )^{2} \)
79$C_2^3$ \( 1 + 11257 T^{2} + 87769968 T^{4} + 11257 p^{4} T^{6} + p^{8} T^{8} \)
83$C_2^2$ \( ( 1 - 13714 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 97 T + 1488 T^{2} - 97 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 88 T + p^{2} T^{2} )^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.608851883695602818545809132685, −8.327002363741599132196297108391, −8.207815022227149660868329732131, −7.977286748951252764029864762324, −7.63852375603967154398646271098, −7.38402988580483885822651657793, −7.29455366833140844899706167295, −7.12262225488378397405422458769, −6.47698602962368446152972701148, −6.27530637637527574808436850450, −6.03548535236801362009784762212, −5.74260047073204042903818080100, −5.29450121193793909041291769494, −5.15518766003113898153590783064, −4.35600323221263858177462550067, −4.09628664832607162624673477700, −3.93120179948246574074877889344, −3.87853562400416745802203871703, −3.53908403006120739696873587713, −3.25500969137098953636602295320, −3.16424864408850206250656724298, −1.75067852966800252895958997178, −1.54673594754687299533832877666, −1.15177743522728845530874508837, −0.06974550543499209033988254261, 0.06974550543499209033988254261, 1.15177743522728845530874508837, 1.54673594754687299533832877666, 1.75067852966800252895958997178, 3.16424864408850206250656724298, 3.25500969137098953636602295320, 3.53908403006120739696873587713, 3.87853562400416745802203871703, 3.93120179948246574074877889344, 4.09628664832607162624673477700, 4.35600323221263858177462550067, 5.15518766003113898153590783064, 5.29450121193793909041291769494, 5.74260047073204042903818080100, 6.03548535236801362009784762212, 6.27530637637527574808436850450, 6.47698602962368446152972701148, 7.12262225488378397405422458769, 7.29455366833140844899706167295, 7.38402988580483885822651657793, 7.63852375603967154398646271098, 7.977286748951252764029864762324, 8.207815022227149660868329732131, 8.327002363741599132196297108391, 8.608851883695602818545809132685

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