Properties

Degree $8$
Conductor $1.642\times 10^{13}$
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

Degree: \(8\)
Conductor: \(3^{4} \cdot 11^{4} \cdot 61^{4}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{2013} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 11^{4} \cdot 61^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4124570705\)
\(L(\frac12)\) \(\approx\) \(0.4124570705\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.65271508152329928500705887755, −6.64392642561724848515497545755, −6.29807843740301208693657585439, −6.22742081900529262251414177567, −5.93556959148486658659893375617, −5.91190183912080268775618973618, −5.77442805733415641430982756458, −5.30296073741477493261327587951, −5.11476482861973461536635262930, −5.00506781233248800967592266795, −4.95478659404725486318513384440, −4.02536232319546232996318940945, −3.90780695540615163050013160301, −3.88778399802942666935190567251, −3.32256207049814413675247257431, −3.12173711911705180404061157220, −2.95776512549937567139967015216, −2.93517158591796632213683886653, −2.93198356080364066099228272194, −1.80312335816194354538799742484, −1.67533993624270204581236008400, −1.51593202432189572704968351758, −1.31947140726701015910000221118, −0.874909967690796563013590971992, −0.867892141232920152992376533936, 0.867892141232920152992376533936, 0.874909967690796563013590971992, 1.31947140726701015910000221118, 1.51593202432189572704968351758, 1.67533993624270204581236008400, 1.80312335816194354538799742484, 2.93198356080364066099228272194, 2.93517158591796632213683886653, 2.95776512549937567139967015216, 3.12173711911705180404061157220, 3.32256207049814413675247257431, 3.88778399802942666935190567251, 3.90780695540615163050013160301, 4.02536232319546232996318940945, 4.95478659404725486318513384440, 5.00506781233248800967592266795, 5.11476482861973461536635262930, 5.30296073741477493261327587951, 5.77442805733415641430982756458, 5.91190183912080268775618973618, 5.93556959148486658659893375617, 6.22742081900529262251414177567, 6.29807843740301208693657585439, 6.64392642561724848515497545755, 6.65271508152329928500705887755

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