Properties

Degree 4
Conductor $ 3^{4} \cdot 19^{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  + 3·4-s + 4·7-s + 6·13-s + 5·16-s + 19-s + 8·25-s + 12·28-s − 10·37-s + 2·49-s + 18·52-s + 16·61-s + 3·64-s − 12·67-s + 4·73-s + 3·76-s − 12·79-s + 24·91-s + 6·97-s + 24·100-s − 4·103-s − 18·109-s + 20·112-s − 4·121-s + 127-s + 131-s + 4·133-s + 137-s + ⋯
L(s)  = 1  + 3/2·4-s + 1.51·7-s + 1.66·13-s + 5/4·16-s + 0.229·19-s + 8/5·25-s + 2.26·28-s − 1.64·37-s + 2/7·49-s + 2.49·52-s + 2.04·61-s + 3/8·64-s − 1.46·67-s + 0.468·73-s + 0.344·76-s − 1.35·79-s + 2.51·91-s + 0.609·97-s + 12/5·100-s − 0.394·103-s − 1.72·109-s + 1.88·112-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.346·133-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(555579\)    =    \(3^{4} \cdot 19^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{555579} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 555579,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $4.139988395$
$L(\frac12)$  $\approx$  $4.139988395$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;19\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
19$C_1$ \( 1 - T \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 36 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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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

−8.408808558589246410485695702154, −7.988450749135605266304232794947, −7.55607400105650970541812714458, −7.04000217879605206023462372692, −6.66762904750664469138144687818, −6.31180265967519847762530496750, −5.69285112016661901477764413384, −5.22311882607786026931767703492, −4.84223630161015872072578480650, −4.05121634773197275874269817884, −3.55399156990710211213919138154, −2.90362156418189127855856742206, −2.31883092052134926433887065330, −1.45940222334492771658126789201, −1.30815728045188892153261341274, 1.30815728045188892153261341274, 1.45940222334492771658126789201, 2.31883092052134926433887065330, 2.90362156418189127855856742206, 3.55399156990710211213919138154, 4.05121634773197275874269817884, 4.84223630161015872072578480650, 5.22311882607786026931767703492, 5.69285112016661901477764413384, 6.31180265967519847762530496750, 6.66762904750664469138144687818, 7.04000217879605206023462372692, 7.55607400105650970541812714458, 7.988450749135605266304232794947, 8.408808558589246410485695702154

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