Properties

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

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.57e3·4-s + 1.16e5·7-s + 1.52e6·13-s − 1.71e6·16-s − 2.06e7·19-s + 2.87e7·25-s − 1.83e8·28-s + 2.12e8·31-s − 1.91e7·37-s + 3.18e9·43-s + 6.17e9·49-s − 2.40e9·52-s − 6.18e9·61-s + 9.30e9·64-s − 1.82e10·67-s + 1.24e9·73-s + 3.24e10·76-s + 2.12e10·79-s + 1.77e11·91-s + 2.63e11·97-s − 4.53e10·100-s + 1.59e11·103-s − 5.78e10·109-s − 1.98e11·112-s − 5.44e11·121-s − 3.34e11·124-s + 127-s + ⋯
L(s)  = 1  − 0.769·4-s + 2.61·7-s + 1.13·13-s − 0.407·16-s − 1.90·19-s + 0.589·25-s − 2.01·28-s + 1.33·31-s − 0.0453·37-s + 3.30·43-s + 3.12·49-s − 0.876·52-s − 0.937·61-s + 1.08·64-s − 1.64·67-s + 0.0700·73-s + 1.46·76-s + 0.776·79-s + 2.97·91-s + 3.11·97-s − 0.453·100-s + 1.35·103-s − 0.360·109-s − 1.06·112-s − 1.90·121-s − 1.02·124-s − 4.98·133-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(81\)    =    \(3^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(11\)
character  :  induced by $\chi_{9} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 81,\ (\ :11/2, 11/2),\ 1)$
$L(6)$  $\approx$  $2.41750$
$L(\frac12)$  $\approx$  $2.41750$
$L(\frac{13}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p(T)\) is a polynomial of degree 4. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
good2$C_2^2$ \( 1 + 197 p^{3} T^{2} + p^{22} T^{4} \)
5$C_2^2$ \( 1 - 5757454 p T^{2} + p^{22} T^{4} \)
7$C_2$ \( ( 1 - 8300 p T + p^{11} T^{2} )^{2} \)
11$C_2^2$ \( 1 + 544818541222 T^{2} + p^{22} T^{4} \)
13$C_2$ \( ( 1 - 762650 T + p^{11} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 11975946694814 T^{2} + p^{22} T^{4} \)
19$C_2$ \( ( 1 + 10301704 T + p^{11} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 1797531507451534 T^{2} + p^{22} T^{4} \)
29$C_2^2$ \( 1 + 4519838782611658 T^{2} + p^{22} T^{4} \)
31$C_2$ \( ( 1 - 106159508 T + p^{11} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 9574450 T + p^{11} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 1088883326741296882 T^{2} + p^{22} T^{4} \)
43$C_2$ \( ( 1 - 1590697400 T + p^{11} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 2869299998598432286 T^{2} + p^{22} T^{4} \)
53$C_2^2$ \( 1 + 17432708152043665114 T^{2} + p^{22} T^{4} \)
59$C_2^2$ \( 1 + 26931896409526885318 T^{2} + p^{22} T^{4} \)
61$C_2$ \( ( 1 + 3092621098 T + p^{11} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 9113820400 T + p^{11} T^{2} )^{2} \)
71$C_2^2$ \( 1 + \)\(45\!\cdots\!42\)\( T^{2} + p^{22} T^{4} \)
73$C_2$ \( ( 1 - 620142950 T + p^{11} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10618486484 T + p^{11} T^{2} )^{2} \)
83$C_2^2$ \( 1 - \)\(10\!\cdots\!46\)\( T^{2} + p^{22} T^{4} \)
89$C_2^2$ \( 1 + \)\(17\!\cdots\!78\)\( T^{2} + p^{22} T^{4} \)
97$C_2$ \( ( 1 - 131872902350 T + p^{11} T^{2} )^{2} \)
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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

−18.67343877165125944631532922126, −18.19580992085115734161371859212, −17.39380097503897935322562441146, −17.32589479212792971417885542288, −15.95126040847141031770763697971, −15.06912800550075981930739313953, −14.41189229539374656407154591115, −13.90728196048111842129494781100, −13.02984707344792266056126019917, −11.95026666745887099259913261182, −10.99426744886364717451432122583, −10.69373083939767882024708034014, −8.944512953201804008681975658185, −8.497689203601718893741955119524, −7.69143782419466166091328575633, −6.08849139898262725735604198615, −4.71439852597627973514333471125, −4.30961620486152328390978910793, −2.09580760927097412764734178325, −0.952851320315113119551955304117, 0.952851320315113119551955304117, 2.09580760927097412764734178325, 4.30961620486152328390978910793, 4.71439852597627973514333471125, 6.08849139898262725735604198615, 7.69143782419466166091328575633, 8.497689203601718893741955119524, 8.944512953201804008681975658185, 10.69373083939767882024708034014, 10.99426744886364717451432122583, 11.95026666745887099259913261182, 13.02984707344792266056126019917, 13.90728196048111842129494781100, 14.41189229539374656407154591115, 15.06912800550075981930739313953, 15.95126040847141031770763697971, 17.32589479212792971417885542288, 17.39380097503897935322562441146, 18.19580992085115734161371859212, 18.67343877165125944631532922126

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