Properties

Degree 4
Conductor $ 3^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 2·4-s + 7-s + 6·9-s + 6·12-s − 9·19-s − 3·21-s + 5·25-s − 9·27-s − 2·28-s + 15·31-s − 12·36-s − 37-s − 10·43-s − 6·49-s + 27·57-s + 12·61-s + 6·63-s + 8·64-s − 11·67-s − 27·73-s − 15·75-s + 18·76-s + 13·79-s + 9·81-s + 6·84-s − 45·93-s + ⋯
L(s)  = 1  − 1.73·3-s − 4-s + 0.377·7-s + 2·9-s + 1.73·12-s − 2.06·19-s − 0.654·21-s + 25-s − 1.73·27-s − 0.377·28-s + 2.69·31-s − 2·36-s − 0.164·37-s − 1.52·43-s − 6/7·49-s + 3.57·57-s + 1.53·61-s + 0.755·63-s + 64-s − 1.34·67-s − 3.16·73-s − 1.73·75-s + 2.06·76-s + 1.46·79-s + 81-s + 0.654·84-s − 4.66·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(441\)    =    \(3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{441} (1, \cdot )$
Sato-Tate  :  $\mathrm{U}(1)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 441,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.2392939029\)
\(L(\frac12)\)  \(\approx\)  \(0.2392939029\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

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

−15.55152607378139954871197988100, −14.86571197707096909151568063946, −14.07800191195924878406018486295, −13.01262643877904661435148088220, −12.97159492547354371966127994182, −11.83685127878360589018873821263, −11.49613855591697476522057219719, −10.37061436856425034139712770950, −10.23518579966480607595970757355, −8.914387133758119867741938583574, −8.219434135148846900962796918633, −6.79345536931461537031444424174, −6.16523471638974682138318287764, −4.87678738065384911829484710659, −4.46412573016302803392609904678, 4.46412573016302803392609904678, 4.87678738065384911829484710659, 6.16523471638974682138318287764, 6.79345536931461537031444424174, 8.219434135148846900962796918633, 8.914387133758119867741938583574, 10.23518579966480607595970757355, 10.37061436856425034139712770950, 11.49613855591697476522057219719, 11.83685127878360589018873821263, 12.97159492547354371966127994182, 13.01262643877904661435148088220, 14.07800191195924878406018486295, 14.86571197707096909151568063946, 15.55152607378139954871197988100

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