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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 3·4-s + 14·7-s + 9·9-s + 12·12-s − 8·13-s + 16·16-s − 32·19-s − 56·21-s − 45·25-s − 44·27-s − 42·28-s + 6·31-s − 27·36-s − 24·37-s + 32·39-s + 176·43-s − 64·48-s + 49·49-s + 24·52-s + 128·57-s + 52·61-s + 126·63-s − 117·64-s − 104·67-s − 36·73-s + 180·75-s + ⋯
L(s)  = 1  − 4/3·3-s − 3/4·4-s + 2·7-s + 9-s + 12-s − 0.615·13-s + 16-s − 1.68·19-s − 8/3·21-s − 9/5·25-s − 1.62·27-s − 3/2·28-s + 6/31·31-s − 3/4·36-s − 0.648·37-s + 0.820·39-s + 4.09·43-s − 4/3·48-s + 49-s + 6/13·52-s + 2.24·57-s + 0.852·61-s + 2·63-s − 1.82·64-s − 1.55·67-s − 0.493·73-s + 12/5·75-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  =  $(8,\ 194481,\ (\ :1, 1, 1, 1),\ 1)$
$L(\frac{3}{2})$  $\approx$  $0.422485$
$L(\frac12)$  $\approx$  $0.422485$
$L(2)$   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$C_2^2$ \( 1 + 4 T + 7 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \)
7$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
good2$C_2^3$ \( 1 + 3 T^{2} - 7 T^{4} + 3 p^{4} T^{6} + p^{8} T^{8} \)
5$C_2^3$ \( 1 + 9 p T^{2} + 56 p^{2} T^{4} + 9 p^{5} T^{6} + p^{8} T^{8} \)
11$C_2^3$ \( 1 + 117 T^{2} - 952 T^{4} + 117 p^{4} T^{6} + p^{8} T^{8} \)
13$C_2$ \( ( 1 + 2 T + p^{2} T^{2} )^{4} \)
17$C_2^3$ \( 1 - 142 T^{2} - 63357 T^{4} - 142 p^{4} T^{6} + p^{8} T^{8} \)
19$C_2^2$ \( ( 1 + 16 T - 105 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$C_2^2$$\times$$C_2^2$ \( ( 1 - 44 T + 1407 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )( 1 + 44 T + 1407 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} ) \)
29$C_2^2$ \( ( 1 - 1437 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 3 T - 952 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 12 T - 1225 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 2382 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 44 T + p^{2} T^{2} )^{4} \)
47$C_2^3$ \( 1 + 4238 T^{2} + 13080963 T^{4} + 4238 p^{4} T^{6} + p^{8} T^{8} \)
53$C_2^3$ \( 1 + 5213 T^{2} + 19284888 T^{4} + 5213 p^{4} T^{6} + p^{8} T^{8} \)
59$C_2^3$ \( 1 + 6557 T^{2} + 30876888 T^{4} + 6557 p^{4} T^{6} + p^{8} T^{8} \)
61$C_2^2$ \( ( 1 - 26 T - 3045 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 52 T - 1785 T^{2} + 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 1262 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 18 T - 5005 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 - p T )^{4}( 1 + p T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 6067 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^3$ \( 1 + 13422 T^{2} + 117407843 T^{4} + 13422 p^{4} T^{6} + p^{8} T^{8} \)
97$C_2$ \( ( 1 + 93 T + p^{2} T^{2} )^{4} \)
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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

−13.78091377295046842559865238585, −13.34860235969718109973705923403, −12.69062168776067391165718117538, −12.56380924719453118065801914145, −12.17389372770494872998564824719, −11.96745805139525695531750575152, −11.36073541210854960998510300557, −11.07786151469147311144936869268, −11.01026287403299148756144258378, −10.47036081354431830917804358796, −10.11171694228189508306715071413, −9.655184331960099459883477600371, −9.193521325281137063349657572869, −8.756563983493013863516331495487, −8.264492582949535754814509523450, −7.73141960301047997538788715575, −7.66361245564424492733683551836, −7.07631093751630237983571049396, −6.13694328109368701907346252175, −5.86317044138694108514974959027, −5.48903860934316606884924599157, −4.77249049080509833323078401705, −4.41658652359578847710526598533, −3.94542690833329969833691836104, −2.02864785952710459585102132239, 2.02864785952710459585102132239, 3.94542690833329969833691836104, 4.41658652359578847710526598533, 4.77249049080509833323078401705, 5.48903860934316606884924599157, 5.86317044138694108514974959027, 6.13694328109368701907346252175, 7.07631093751630237983571049396, 7.66361245564424492733683551836, 7.73141960301047997538788715575, 8.264492582949535754814509523450, 8.756563983493013863516331495487, 9.193521325281137063349657572869, 9.655184331960099459883477600371, 10.11171694228189508306715071413, 10.47036081354431830917804358796, 11.01026287403299148756144258378, 11.07786151469147311144936869268, 11.36073541210854960998510300557, 11.96745805139525695531750575152, 12.17389372770494872998564824719, 12.56380924719453118065801914145, 12.69062168776067391165718117538, 13.34860235969718109973705923403, 13.78091377295046842559865238585

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