Properties

Degree 4
Conductor $ 3^{2} \cdot 37^{2} \cdot 73 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4-s + 2·7-s + 9-s − 2·12-s − 3·16-s − 4·21-s − 2·25-s + 4·27-s + 2·28-s − 6·31-s + 36-s + 2·37-s − 6·43-s + 6·48-s − 2·49-s + 12·61-s + 2·63-s − 7·64-s − 4·67-s − 3·73-s + 4·75-s + 24·79-s − 11·81-s − 4·84-s + 12·93-s − 12·97-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s − 0.577·12-s − 3/4·16-s − 0.872·21-s − 2/5·25-s + 0.769·27-s + 0.377·28-s − 1.07·31-s + 1/6·36-s + 0.328·37-s − 0.914·43-s + 0.866·48-s − 2/7·49-s + 1.53·61-s + 0.251·63-s − 7/8·64-s − 0.488·67-s − 0.351·73-s + 0.461·75-s + 2.70·79-s − 1.22·81-s − 0.436·84-s + 1.24·93-s − 1.21·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 899433 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 899433 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(899433\)    =    \(3^{2} \cdot 37^{2} \cdot 73\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{899433} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 899433,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;37,\;73\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;37,\;73\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( 1 + 2 T + p T^{2} \)
37$C_2$ \( 1 - 2 T + p T^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−7.86330777758687142709490070726, −7.44608718050185272124904712398, −7.01110247650939255265602033279, −6.55119379829724561772210766439, −6.21336511106571555768373070773, −5.73503107641816970806493434862, −5.16237715118046192009070636270, −4.97395715255386048392047123858, −4.39333420742110610326234484876, −3.79781138113193643566883087183, −3.18946674071368299697675385549, −2.36320011203475387280512143988, −1.91701188221127242969254448131, −1.08090988097063989704221367766, 0, 1.08090988097063989704221367766, 1.91701188221127242969254448131, 2.36320011203475387280512143988, 3.18946674071368299697675385549, 3.79781138113193643566883087183, 4.39333420742110610326234484876, 4.97395715255386048392047123858, 5.16237715118046192009070636270, 5.73503107641816970806493434862, 6.21336511106571555768373070773, 6.55119379829724561772210766439, 7.01110247650939255265602033279, 7.44608718050185272124904712398, 7.86330777758687142709490070726

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