Properties

Degree 6
Conductor $ 2011^{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  + 3·4-s − 5-s + 3·9-s − 13-s + 6·16-s − 3·20-s − 23-s − 31-s + 9·36-s − 41-s − 43-s − 3·45-s + 3·49-s − 3·52-s + 10·64-s + 65-s − 71-s − 6·80-s + 6·81-s − 83-s − 89-s − 3·92-s − 101-s − 103-s − 109-s + 115-s − 3·117-s + ⋯
L(s)  = 1  + 3·4-s − 5-s + 3·9-s − 13-s + 6·16-s − 3·20-s − 23-s − 31-s + 9·36-s − 41-s − 43-s − 3·45-s + 3·49-s − 3·52-s + 10·64-s + 65-s − 71-s − 6·80-s + 6·81-s − 83-s − 89-s − 3·92-s − 101-s − 103-s − 109-s + 115-s − 3·117-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2011^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2011^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(2011^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{2011} (2010, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(6,\ 2011^{3} ,\ ( \ : 0, 0, 0 ),\ 1 )$
$L(\frac{1}{2})$  $\approx$  $3.179218014$
$L(\frac12)$  $\approx$  $3.179218014$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2011$, \(F_p\) is a polynomial of degree 6. If $p = 2011$, then $F_p$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p$
bad2011 \( 1+O(T) \)
good2$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
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$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
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$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
41$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
43$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
47$C_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$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
71$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
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$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
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

−8.260296202951996999396743795541, −7.71034924230206571734003232675, −7.64499294080361593415047298585, −7.45881190023126099539701954613, −7.13914371461768728472912077340, −7.05296604711855684608828411781, −7.03384206160308353072004579797, −6.60257240899894917415705156396, −6.25947800722014296123538162185, −6.14374611519011938166547855601, −5.63800325129470412524547640051, −5.41764417430903220491527889275, −5.17284836000195726489118549613, −4.65219973318233909750970594027, −4.34831431540597383193760842166, −4.07556281147085956434886199465, −3.75370341881959017979953493370, −3.52165873841477642791745100310, −3.28691773354206112475033510574, −2.64131041767919126497260176640, −2.34543493833801924716716633503, −2.21394021840978122774154295075, −1.67988930515697396102481732042, −1.40324453744509293915956313039, −1.14161997012900792784024894394, 1.14161997012900792784024894394, 1.40324453744509293915956313039, 1.67988930515697396102481732042, 2.21394021840978122774154295075, 2.34543493833801924716716633503, 2.64131041767919126497260176640, 3.28691773354206112475033510574, 3.52165873841477642791745100310, 3.75370341881959017979953493370, 4.07556281147085956434886199465, 4.34831431540597383193760842166, 4.65219973318233909750970594027, 5.17284836000195726489118549613, 5.41764417430903220491527889275, 5.63800325129470412524547640051, 6.14374611519011938166547855601, 6.25947800722014296123538162185, 6.60257240899894917415705156396, 7.03384206160308353072004579797, 7.05296604711855684608828411781, 7.13914371461768728472912077340, 7.45881190023126099539701954613, 7.64499294080361593415047298585, 7.71034924230206571734003232675, 8.260296202951996999396743795541

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