Properties

Degree 4
Conductor $ 3^{3} \cdot 5 \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4-s − 5-s + 9-s + 6·11-s − 12-s − 15-s − 3·16-s + 20-s + 12·23-s − 4·25-s + 27-s + 4·31-s + 6·33-s − 36-s − 8·37-s − 6·44-s − 45-s − 3·48-s + 2·49-s + 6·53-s − 6·55-s − 12·59-s + 60-s + 7·64-s − 8·67-s + 12·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 1/2·4-s − 0.447·5-s + 1/3·9-s + 1.80·11-s − 0.288·12-s − 0.258·15-s − 3/4·16-s + 0.223·20-s + 2.50·23-s − 4/5·25-s + 0.192·27-s + 0.718·31-s + 1.04·33-s − 1/6·36-s − 1.31·37-s − 0.904·44-s − 0.149·45-s − 0.433·48-s + 2/7·49-s + 0.824·53-s − 0.809·55-s − 1.56·59-s + 0.129·60-s + 7/8·64-s − 0.977·67-s + 1.44·69-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16335 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16335 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(16335\)    =    \(3^{3} \cdot 5 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{16335} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 16335,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.199237719$
$L(\frac12)$  $\approx$  $1.199237719$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$ \( 1 - T \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
11$C_2$ \( 1 - 6 T + p T^{2} \)
good2$V_4$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$V_4$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$V_4$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$V_4$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
43$V_4$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$V_4$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
83$V_4$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−11.07885762480374341503699542688, −10.53877703244282914988870891494, −9.801642233543017658171021786499, −9.184779578408665051933646655609, −8.913758611637897504117510519972, −8.587908174628606286567997361203, −7.66666527884911410422785883674, −7.06406041849727788604476099438, −6.69984433536507155425208888227, −5.86349685195503580982602479683, −4.86185008294020342641002909596, −4.35581010965663593455890117092, −3.66814240836412844247219877757, −2.88267927841335234319026718021, −1.43632818097350117902865692288, 1.43632818097350117902865692288, 2.88267927841335234319026718021, 3.66814240836412844247219877757, 4.35581010965663593455890117092, 4.86185008294020342641002909596, 5.86349685195503580982602479683, 6.69984433536507155425208888227, 7.06406041849727788604476099438, 7.66666527884911410422785883674, 8.587908174628606286567997361203, 8.913758611637897504117510519972, 9.184779578408665051933646655609, 9.801642233543017658171021786499, 10.53877703244282914988870891494, 11.07885762480374341503699542688

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