Properties

Degree 6
Conductor $ 3^{6} \cdot 7^{3} \cdot 41^{3} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3·7-s − 13-s + 3·14-s + 17-s − 19-s + 23-s + 3·25-s − 26-s + 34-s − 37-s − 38-s − 3·41-s − 43-s + 46-s + 47-s + 6·49-s + 3·50-s − 74-s − 3·82-s − 86-s + 89-s − 3·91-s + 94-s − 97-s + 6·98-s + 101-s + ⋯
L(s)  = 1  + 2-s + 3·7-s − 13-s + 3·14-s + 17-s − 19-s + 23-s + 3·25-s − 26-s + 34-s − 37-s − 38-s − 3·41-s − 43-s + 46-s + 47-s + 6·49-s + 3·50-s − 74-s − 3·82-s − 86-s + 89-s − 3·91-s + 94-s − 97-s + 6·98-s + 101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{3} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{3} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(3^{6} \cdot 7^{3} \cdot 41^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{2583} (2008, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((6,\ 3^{6} \cdot 7^{3} \cdot 41^{3} ,\ ( \ : 0, 0, 0 ),\ 1 )\)
\(L(\frac{1}{2})\)  \(\approx\)  \(3.722861775\)
\(L(\frac12)\)  \(\approx\)  \(3.722861775\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;41\}$,\(F_p(T)\) is a polynomial of degree 6. If $p \in \{3,\;7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
7$C_1$ \( ( 1 - T )^{3} \)
41$C_1$ \( ( 1 + T )^{3} \)
good2$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
13$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
17$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
19$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
23$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
37$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
43$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
47$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
89$C_6$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
97$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.976857762952515399555409853895, −7.932871141518870236024695316004, −7.48567206600870532646318121517, −7.28914788626878081825581172147, −7.00445360579493329952128281593, −6.85861175945590103652738174124, −6.66009108189749755568573142615, −6.19050079647528689482827761777, −5.68550212017814137575769996683, −5.59944926380598088895231367307, −5.13319449503001339350485020187, −5.05962152005061891764338708651, −4.93474610212934861695038140877, −4.60565933895355423887421195527, −4.48618270099382737373914540532, −4.36997224808589748301786221014, −3.59931104148218534627880623277, −3.48442627967473670769823122692, −3.27640570177130211155257930653, −2.54489260032120497780155444424, −2.54189142277243340413242525366, −1.86169295457560911973671366546, −1.83975269323596061561585091536, −1.09267674343460805980299570314, −1.09225969864206342020923818810, 1.09225969864206342020923818810, 1.09267674343460805980299570314, 1.83975269323596061561585091536, 1.86169295457560911973671366546, 2.54189142277243340413242525366, 2.54489260032120497780155444424, 3.27640570177130211155257930653, 3.48442627967473670769823122692, 3.59931104148218534627880623277, 4.36997224808589748301786221014, 4.48618270099382737373914540532, 4.60565933895355423887421195527, 4.93474610212934861695038140877, 5.05962152005061891764338708651, 5.13319449503001339350485020187, 5.59944926380598088895231367307, 5.68550212017814137575769996683, 6.19050079647528689482827761777, 6.66009108189749755568573142615, 6.85861175945590103652738174124, 7.00445360579493329952128281593, 7.28914788626878081825581172147, 7.48567206600870532646318121517, 7.932871141518870236024695316004, 7.976857762952515399555409853895

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