Properties

Degree 4
Conductor $ 3^{2} \cdot 11^{2} \cdot 17^{2} $
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  + 2·3-s − 4·4-s + 6·5-s + 3·9-s − 3·11-s − 8·12-s + 12·15-s + 12·16-s − 24·20-s + 18·23-s + 17·25-s + 4·27-s + 4·31-s − 6·33-s − 12·36-s − 8·37-s + 12·44-s + 18·45-s − 12·47-s + 24·48-s + 2·49-s − 12·53-s − 18·55-s + 12·59-s − 48·60-s − 32·64-s − 8·67-s + ⋯
L(s)  = 1  + 1.15·3-s − 2·4-s + 2.68·5-s + 9-s − 0.904·11-s − 2.30·12-s + 3.09·15-s + 3·16-s − 5.36·20-s + 3.75·23-s + 17/5·25-s + 0.769·27-s + 0.718·31-s − 1.04·33-s − 2·36-s − 1.31·37-s + 1.80·44-s + 2.68·45-s − 1.75·47-s + 3.46·48-s + 2/7·49-s − 1.64·53-s − 2.42·55-s + 1.56·59-s − 6.19·60-s − 4·64-s − 0.977·67-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 314721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 314721 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(314721\)    =    \(3^{2} \cdot 11^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{314721} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 314721,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $3.038180991$
$L(\frac12)$  $\approx$  $3.038180991$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;11,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$ \( ( 1 - T )^{2} \)
11$C_2$ \( 1 + 3 T + p T^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.873138966846700744402216412888, −8.618966621189813626434174913816, −8.123063877468170217097057069948, −7.60195787768679259689608146982, −6.66463575826783700250676263362, −6.65141393178900011874126957740, −5.62535288746366565403300368914, −5.18754379686639968375671320234, −5.14306131647545403565477997005, −4.59411129964790589621534187851, −3.66589986463958216194059753670, −3.00632648437536992753290178800, −2.69595897502735310420716823107, −1.70457407581975008871008195419, −1.09494185617481664600047301525, 1.09494185617481664600047301525, 1.70457407581975008871008195419, 2.69595897502735310420716823107, 3.00632648437536992753290178800, 3.66589986463958216194059753670, 4.59411129964790589621534187851, 5.14306131647545403565477997005, 5.18754379686639968375671320234, 5.62535288746366565403300368914, 6.65141393178900011874126957740, 6.66463575826783700250676263362, 7.60195787768679259689608146982, 8.123063877468170217097057069948, 8.618966621189813626434174913816, 8.873138966846700744402216412888

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