Properties

Degree 4
Conductor $ 3^{2} \cdot 7^{2} \cdot 11^{3} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·3-s − 4-s + 4·6-s − 8·8-s + 9-s + 11-s − 2·12-s − 7·16-s + 8·17-s + 2·18-s + 2·22-s − 16·24-s − 6·25-s − 4·27-s − 12·29-s + 20·31-s + 14·32-s + 2·33-s + 16·34-s − 36-s − 12·37-s + 8·41-s − 44-s − 14·48-s + 49-s − 12·50-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s − 1/2·4-s + 1.63·6-s − 2.82·8-s + 1/3·9-s + 0.301·11-s − 0.577·12-s − 7/4·16-s + 1.94·17-s + 0.471·18-s + 0.426·22-s − 3.26·24-s − 6/5·25-s − 0.769·27-s − 2.22·29-s + 3.59·31-s + 2.47·32-s + 0.348·33-s + 2.74·34-s − 1/6·36-s − 1.97·37-s + 1.24·41-s − 0.150·44-s − 2.02·48-s + 1/7·49-s − 1.69·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(586971\)    =    \(3^{2} \cdot 7^{2} \cdot 11^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{586971} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 586971,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.671852040\)
\(L(\frac12)\)  \(\approx\)  \(3.671852040\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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\[\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.576763002125084546471923487790, −7.87739468442856243522484593458, −7.82867506007281568614416512666, −7.07007000345216400828661927108, −6.31251542982324707519501957045, −5.96443978110022033983848919969, −5.45217811977635791883277118514, −5.22894511431647767966156393918, −4.45772251821753336888632495633, −4.07930883034575373640745113861, −3.58922471493276128945836920884, −3.26888344602981049454952325554, −2.78349816157540378309057592549, −1.96167877918167104055842099400, −0.75006805308184274174858297412, 0.75006805308184274174858297412, 1.96167877918167104055842099400, 2.78349816157540378309057592549, 3.26888344602981049454952325554, 3.58922471493276128945836920884, 4.07930883034575373640745113861, 4.45772251821753336888632495633, 5.22894511431647767966156393918, 5.45217811977635791883277118514, 5.96443978110022033983848919969, 6.31251542982324707519501957045, 7.07007000345216400828661927108, 7.82867506007281568614416512666, 7.87739468442856243522484593458, 8.576763002125084546471923487790

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