Properties

Degree 8
Conductor $ 3^{4} \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  − 2·3-s + 2·4-s − 8·9-s − 4·12-s − 36·13-s − 16-s + 12·19-s + 44·25-s + 22·27-s + 136·31-s − 16·36-s + 16·37-s + 72·39-s − 160·43-s + 2·48-s + 14·49-s − 72·52-s − 24·57-s − 156·61-s + 20·64-s − 24·67-s − 32·73-s − 88·75-s + 24·76-s + 128·79-s + 7·81-s − 272·93-s + ⋯
L(s)  = 1  − 2/3·3-s + 1/2·4-s − 8/9·9-s − 1/3·12-s − 2.76·13-s − 0.0625·16-s + 0.631·19-s + 1.75·25-s + 0.814·27-s + 4.38·31-s − 4/9·36-s + 0.432·37-s + 1.84·39-s − 3.72·43-s + 1/24·48-s + 2/7·49-s − 1.38·52-s − 0.421·57-s − 2.55·61-s + 5/16·64-s − 0.358·67-s − 0.438·73-s − 1.17·75-s + 6/19·76-s + 1.62·79-s + 7/81·81-s − 2.92·93-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(194481\)    =    \(3^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{21} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 194481,\ (\ :1, 1, 1, 1),\ 1)$
$L(\frac{3}{2})$  $\approx$  $0.511509$
$L(\frac12)$  $\approx$  $0.511509$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7\}$, \(F_p\) is a polynomial of degree 8. If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad3$D_{4}$ \( 1 + 2 T + 4 p T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
good2$C_2^2 \wr C_2$ \( 1 - p T^{2} + 5 T^{4} - p^{5} T^{6} + p^{8} T^{8} \)
5$C_2^2 \wr C_2$ \( 1 - 44 T^{2} + 1034 T^{4} - 44 p^{4} T^{6} + p^{8} T^{8} \)
11$C_2^2 \wr C_2$ \( 1 - 428 T^{2} + 74630 T^{4} - 428 p^{4} T^{6} + p^{8} T^{8} \)
13$D_{4}$ \( ( 1 + 18 T + 412 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
17$C_2^2 \wr C_2$ \( 1 - 988 T^{2} + 407046 T^{4} - 988 p^{4} T^{6} + p^{8} T^{8} \)
19$D_{4}$ \( ( 1 - 6 T + 556 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - 1444 T^{2} + 980166 T^{4} - 1444 p^{4} T^{6} + p^{8} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 - 2972 T^{2} + 3611558 T^{4} - 2972 p^{4} T^{6} + p^{8} T^{8} \)
31$D_{4}$ \( ( 1 - 68 T + 3050 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 8 T + 1382 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - 1292 T^{2} + 2832038 T^{4} - 1292 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 + 80 T + 5046 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 - 6148 T^{2} + 19144326 T^{4} - 6148 p^{4} T^{6} + p^{8} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 + 20 T^{2} - 13350138 T^{4} + 20 p^{4} T^{6} + p^{8} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 - 3676 T^{2} + 15964266 T^{4} - 3676 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 + 78 T + 8620 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 12 T + 8566 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 - 10588 T^{2} + 78813510 T^{4} - 10588 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 + 16 T + 5990 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 64 T + 4434 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 - 13948 T^{2} + 141899946 T^{4} - 13948 p^{4} T^{6} + p^{8} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 - 11468 T^{2} + 120945830 T^{4} - 11468 p^{4} T^{6} + p^{8} T^{8} \)
97$D_{4}$ \( ( 1 + 4 T + 18374 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−13.70199429714925516254534406272, −13.57754343007456276063292812852, −12.77676763769673097375325456987, −12.21212455621933931337508124739, −12.15326923744696763363963554493, −12.05215955094747283959721173692, −11.61055449222500752374842862982, −11.18926705230815417960428913541, −10.89543734001262305707354309184, −10.22044971433123207637602782910, −9.951468482733200695346432232765, −9.912622864366688966299283425911, −9.295502806591169357036349258163, −8.602616431874748734043358950999, −8.365753564601864572636346836896, −7.890427583737138323944897633325, −7.31850106587427905497581587199, −6.94700266496420815955978361283, −6.35514300323772687842844043143, −6.23009728629170269715395110239, −5.21070435478957792899505003238, −4.76339774791315908076859192989, −4.76189951599963393312237247570, −2.99522710624722766591794103112, −2.72394693730293101925304774111, 2.72394693730293101925304774111, 2.99522710624722766591794103112, 4.76189951599963393312237247570, 4.76339774791315908076859192989, 5.21070435478957792899505003238, 6.23009728629170269715395110239, 6.35514300323772687842844043143, 6.94700266496420815955978361283, 7.31850106587427905497581587199, 7.890427583737138323944897633325, 8.365753564601864572636346836896, 8.602616431874748734043358950999, 9.295502806591169357036349258163, 9.912622864366688966299283425911, 9.951468482733200695346432232765, 10.22044971433123207637602782910, 10.89543734001262305707354309184, 11.18926705230815417960428913541, 11.61055449222500752374842862982, 12.05215955094747283959721173692, 12.15326923744696763363963554493, 12.21212455621933931337508124739, 12.77676763769673097375325456987, 13.57754343007456276063292812852, 13.70199429714925516254534406272

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