Properties

Degree 4
Conductor $ 3^{2} \cdot 37^{2} $
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-s − 4-s + 7-s + 12-s + 13-s − 2·19-s − 21-s − 25-s + 27-s − 28-s − 2·31-s + 2·37-s − 39-s − 2·43-s + 49-s − 52-s + 2·57-s − 2·61-s + 64-s + 67-s − 2·73-s + 75-s + 2·76-s + 79-s − 81-s + 84-s + 91-s + ⋯
L(s)  = 1  − 3-s − 4-s + 7-s + 12-s + 13-s − 2·19-s − 21-s − 25-s + 27-s − 28-s − 2·31-s + 2·37-s − 39-s − 2·43-s + 49-s − 52-s + 2·57-s − 2·61-s + 64-s + 67-s − 2·73-s + 75-s + 2·76-s + 79-s − 81-s + 84-s + 91-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(12321\)    =    \(3^{2} \cdot 37^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{111} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 12321,\ (\ :0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.2374574843$
$L(\frac12)$  $\approx$  $0.2374574843$
$L(1)$   not available
$L(1)$   not available

Euler product

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

−14.08859395282092172934470765459, −13.62304570253849326090067990079, −12.93352522765451202314856167685, −12.87351548979488832581956316560, −11.95902905251912013524634114598, −11.50359384581484360434866422988, −10.96748126909624518403723367049, −10.80389228668928613087785431902, −10.00763679155022500534844766937, −9.323550309234632306437585417334, −8.616652078435595282895870814145, −8.463172696586380663441779432537, −7.70723203883669501750919805383, −6.86328204660327443830310453966, −5.97703246902699662443433916656, −5.81220143111909919666559615863, −4.74283713726553977917887541470, −4.48691441166618421750396295135, −3.60018671529121378340731000668, −1.94869000785363089009706569365, 1.94869000785363089009706569365, 3.60018671529121378340731000668, 4.48691441166618421750396295135, 4.74283713726553977917887541470, 5.81220143111909919666559615863, 5.97703246902699662443433916656, 6.86328204660327443830310453966, 7.70723203883669501750919805383, 8.463172696586380663441779432537, 8.616652078435595282895870814145, 9.323550309234632306437585417334, 10.00763679155022500534844766937, 10.80389228668928613087785431902, 10.96748126909624518403723367049, 11.50359384581484360434866422988, 11.95902905251912013524634114598, 12.87351548979488832581956316560, 12.93352522765451202314856167685, 13.62304570253849326090067990079, 14.08859395282092172934470765459

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