Properties

Degree 8
Conductor $ 3^{4} \cdot 11^{4} \cdot 61^{4} $
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  + 3·2-s − 3-s + 6·4-s − 3·6-s + 10·8-s − 11-s − 6·12-s + 3·13-s + 15·16-s − 2·17-s + 3·19-s − 3·22-s − 2·23-s − 10·24-s − 25-s + 9·26-s − 2·29-s + 22·32-s + 33-s − 6·34-s + 9·38-s − 3·39-s − 6·44-s − 6·46-s − 15·48-s − 49-s − 3·50-s + ⋯
L(s)  = 1  + 3·2-s − 3-s + 6·4-s − 3·6-s + 10·8-s − 11-s − 6·12-s + 3·13-s + 15·16-s − 2·17-s + 3·19-s − 3·22-s − 2·23-s − 10·24-s − 25-s + 9·26-s − 2·29-s + 22·32-s + 33-s − 6·34-s + 9·38-s − 3·39-s − 6·44-s − 6·46-s − 15·48-s − 49-s − 3·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{4} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{4} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(3^{4} \cdot 11^{4} \cdot 61^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{2013} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 3^{4} \cdot 11^{4} \cdot 61^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)
\(L(\frac{1}{2})\)  \(\approx\)  \(8.691883791\)
\(L(\frac12)\)  \(\approx\)  \(8.691883791\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11,\;61\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{3,\;11,\;61\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
11$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
61$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
good2$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
5$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
7$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
17$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
19$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
23$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
29$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
41$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
53$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
59$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
71$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
73$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
79$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
89$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
97$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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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

−6.43063132910694996140041416001, −6.33566622316266720514617648492, −6.20159755150888266028503701119, −5.81820661742036333616827116408, −5.78026527202102135929968583138, −5.59971976181835431385930481035, −5.56919691033676277124904450063, −5.22468041487568057483296394102, −5.19702476317934490884801871754, −5.01299527421013749686078410533, −4.29358346710920206862928083509, −4.28606516916897942383683105997, −4.11513672882863889336920846109, −3.95289697488769425724936961715, −3.89692965906837451583909724345, −3.59221123046221468534299891148, −3.24588558521075082405944708887, −2.87667747755197645752423453914, −2.75485978529463267966960097434, −2.64410846358512875662186522678, −2.25098831899809015604863774115, −1.68886661510674863768926731397, −1.57544670175381383041692720923, −1.51168696704987709529697247973, −0.902967577569815416861286500863, 0.902967577569815416861286500863, 1.51168696704987709529697247973, 1.57544670175381383041692720923, 1.68886661510674863768926731397, 2.25098831899809015604863774115, 2.64410846358512875662186522678, 2.75485978529463267966960097434, 2.87667747755197645752423453914, 3.24588558521075082405944708887, 3.59221123046221468534299891148, 3.89692965906837451583909724345, 3.95289697488769425724936961715, 4.11513672882863889336920846109, 4.28606516916897942383683105997, 4.29358346710920206862928083509, 5.01299527421013749686078410533, 5.19702476317934490884801871754, 5.22468041487568057483296394102, 5.56919691033676277124904450063, 5.59971976181835431385930481035, 5.78026527202102135929968583138, 5.81820661742036333616827116408, 6.20159755150888266028503701119, 6.33566622316266720514617648492, 6.43063132910694996140041416001

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