Properties

Degree 6
Conductor $ 71^{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-s − 5-s + 6-s + 10-s + 15-s − 19-s − 29-s − 30-s − 37-s + 38-s − 43-s + 3·49-s + 57-s + 58-s + 3·71-s − 73-s + 74-s − 79-s − 83-s + 86-s + 87-s − 89-s + 95-s − 3·98-s − 101-s − 103-s + ⋯
L(s)  = 1  − 2-s − 3-s − 5-s + 6-s + 10-s + 15-s − 19-s − 29-s − 30-s − 37-s + 38-s − 43-s + 3·49-s + 57-s + 58-s + 3·71-s − 73-s + 74-s − 79-s − 83-s + 86-s + 87-s − 89-s + 95-s − 3·98-s − 101-s − 103-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 357911 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 357911 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(357911\)    =    \(71^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{71} (70, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(6,\ 357911,\ (\ :0, 0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.05498223480$
$L(\frac12)$  $\approx$  $0.05498223480$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 71$, \(F_p\) is a polynomial of degree 6. If $p = 71$, then $F_p$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p$
bad71$C_1$ \( ( 1 - T )^{3} \)
good2$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
3$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
5$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
7$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_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
19$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
29$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
37$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
43$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
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$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
73$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
79$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
83$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
89$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
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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

−13.58399519906890803499351778176, −12.88778415189161766638407934767, −12.62842911673974380337261507666, −12.42351194541329992426458232503, −11.75493220562099759167484344236, −11.67985104016781996899636064300, −11.17081547941659221637655464575, −11.05204135964904542931077288526, −10.37263061263202496602275828991, −10.24785144415084892602116520791, −9.646341880691708918339787590058, −9.249596659145116242975209696115, −8.775621347957351956553642935712, −8.392664296558968410521780245040, −8.308375254782647564178629435651, −7.34866153009380292102541584258, −7.32935458632589252277476848657, −6.76877928720481535930982673330, −6.02203245587172561884189386059, −5.76082408647750230290643880271, −5.19361966854786324344583978535, −4.51844117616935213608968885293, −3.95531641366854447656333489742, −3.39979849702102685652285544820, −2.19448912992332732423794638884, 2.19448912992332732423794638884, 3.39979849702102685652285544820, 3.95531641366854447656333489742, 4.51844117616935213608968885293, 5.19361966854786324344583978535, 5.76082408647750230290643880271, 6.02203245587172561884189386059, 6.76877928720481535930982673330, 7.32935458632589252277476848657, 7.34866153009380292102541584258, 8.308375254782647564178629435651, 8.392664296558968410521780245040, 8.775621347957351956553642935712, 9.249596659145116242975209696115, 9.646341880691708918339787590058, 10.24785144415084892602116520791, 10.37263061263202496602275828991, 11.05204135964904542931077288526, 11.17081547941659221637655464575, 11.67985104016781996899636064300, 11.75493220562099759167484344236, 12.42351194541329992426458232503, 12.62842911673974380337261507666, 12.88778415189161766638407934767, 13.58399519906890803499351778176

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