Properties

Degree 6
Conductor $ 2^{9} \cdot 251^{3} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 6·4-s − 7-s + 10·8-s + 3·9-s − 11-s − 3·14-s + 15·16-s − 17-s + 9·18-s − 19-s − 3·22-s − 23-s + 3·25-s − 6·28-s − 29-s − 31-s + 21·32-s − 3·34-s + 18·36-s − 37-s − 3·38-s − 41-s − 43-s − 6·44-s − 3·46-s + 9·50-s + ⋯
L(s)  = 1  + 3·2-s + 6·4-s − 7-s + 10·8-s + 3·9-s − 11-s − 3·14-s + 15·16-s − 17-s + 9·18-s − 19-s − 3·22-s − 23-s + 3·25-s − 6·28-s − 29-s − 31-s + 21·32-s − 3·34-s + 18·36-s − 37-s − 3·38-s − 41-s − 43-s − 6·44-s − 3·46-s + 9·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 251^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 251^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(6\)
\( N \)  =  \(2^{9} \cdot 251^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{2008} (501, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((6,\ 2^{9} \cdot 251^{3} ,\ ( \ : 0, 0, 0 ),\ 1 )\)
\(L(\frac{1}{2})\)  \(\approx\)  \(11.31177397\)
\(L(\frac12)\)  \(\approx\)  \(11.31177397\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;251\}$,\(F_p(T)\) is a polynomial of degree 6. If $p \in \{2,\;251\}$, then $F_p(T)$ is a polynomial of degree at most 5.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - T )^{3} \)
251$C_1$ \( ( 1 - T )^{3} \)
good3$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
7$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
11$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{3}( 1 + T )^{3} \)
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$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
31$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
37$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
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$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
59$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
61$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
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$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
79$C_6$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \)
83$C_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_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

−8.072614514188752656720750219597, −7.82135070647735025724817902482, −7.57219883163814118285651831258, −7.18967309368170271650745755970, −6.96286850243061467356775829999, −6.85796850160243507272025522402, −6.85446611413567599809651854315, −6.35289723487539999629854009989, −6.10152828771663925452510106786, −6.09730481407786616323716315548, −5.31893453327012316495797516672, −5.13453697736513187885014052533, −5.13081514742221194470586310891, −4.68071065382005982257663483019, −4.40157628475435466194178330581, −4.26276995996199409891652902832, −3.93671003577164193694341772594, −3.63280360133122603551530594142, −3.38788895230886524496922558945, −2.85265783201785653389638142199, −2.83085693221179326680520788488, −2.30140953731774952545730097360, −1.79916743201281788443203136104, −1.51044338409629372437500213854, −1.46625051177396352223191823842, 1.46625051177396352223191823842, 1.51044338409629372437500213854, 1.79916743201281788443203136104, 2.30140953731774952545730097360, 2.83085693221179326680520788488, 2.85265783201785653389638142199, 3.38788895230886524496922558945, 3.63280360133122603551530594142, 3.93671003577164193694341772594, 4.26276995996199409891652902832, 4.40157628475435466194178330581, 4.68071065382005982257663483019, 5.13081514742221194470586310891, 5.13453697736513187885014052533, 5.31893453327012316495797516672, 6.09730481407786616323716315548, 6.10152828771663925452510106786, 6.35289723487539999629854009989, 6.85446611413567599809651854315, 6.85796850160243507272025522402, 6.96286850243061467356775829999, 7.18967309368170271650745755970, 7.57219883163814118285651831258, 7.82135070647735025724817902482, 8.072614514188752656720750219597

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