Properties

Degree 4
Conductor $ 3^{2} \cdot 11^{3} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 3·4-s − 4·5-s + 9-s + 11-s + 6·12-s − 8·15-s + 5·16-s − 12·20-s + 4·23-s + 2·25-s − 4·27-s + 2·31-s + 2·33-s + 3·36-s − 12·37-s + 3·44-s − 4·45-s − 2·47-s + 10·48-s + 8·49-s − 6·53-s − 4·55-s − 8·59-s − 24·60-s + 3·64-s + 10·67-s + ⋯
L(s)  = 1  + 1.15·3-s + 3/2·4-s − 1.78·5-s + 1/3·9-s + 0.301·11-s + 1.73·12-s − 2.06·15-s + 5/4·16-s − 2.68·20-s + 0.834·23-s + 2/5·25-s − 0.769·27-s + 0.359·31-s + 0.348·33-s + 1/2·36-s − 1.97·37-s + 0.452·44-s − 0.596·45-s − 0.291·47-s + 1.44·48-s + 8/7·49-s − 0.824·53-s − 0.539·55-s − 1.04·59-s − 3.09·60-s + 3/8·64-s + 1.22·67-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 11979 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 11979 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(11979\)    =    \(3^{2} \cdot 11^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{11979} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 11979,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.478974579$
$L(\frac12)$  $\approx$  $1.478974579$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_2$ \( 1 - 2 T + p T^{2} \)
11$C_1$ \( 1 - T \)
good2$V_4$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$V_4$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
13$V_4$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
17$V_4$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
19$V_4$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$V_4$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$V_4$ \( 1 + 68 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$V_4$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
79$V_4$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
83$V_4$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−11.45131824258061458916450519696, −10.91690577990793610660900803282, −10.45579235275946954710273984552, −9.582086136330591251915885065607, −8.977148289736582097082004206859, −8.275820978254645125631912579542, −7.969960679393019470039862150512, −7.35870763525177012678764393491, −6.99849377937138192702490611689, −6.28826950708592892337652477350, −5.30030269977841569591804308880, −4.19875107801205039988739345512, −3.52202012479345836653568205274, −3.00153015912170044874275189793, −1.95180531308691071472655104039, 1.95180531308691071472655104039, 3.00153015912170044874275189793, 3.52202012479345836653568205274, 4.19875107801205039988739345512, 5.30030269977841569591804308880, 6.28826950708592892337652477350, 6.99849377937138192702490611689, 7.35870763525177012678764393491, 7.969960679393019470039862150512, 8.275820978254645125631912579542, 8.977148289736582097082004206859, 9.582086136330591251915885065607, 10.45579235275946954710273984552, 10.91690577990793610660900803282, 11.45131824258061458916450519696

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