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  − 4-s + 2·7-s − 6·13-s − 3·16-s − 19-s + 4·25-s − 2·28-s − 6·31-s + 4·43-s − 2·49-s + 6·52-s + 8·61-s + 7·64-s + 6·67-s + 4·73-s + 76-s + 12·79-s − 12·91-s − 18·97-s − 4·100-s + 24·103-s − 18·109-s − 6·112-s − 14·121-s + 6·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.755·7-s − 1.66·13-s − 3/4·16-s − 0.229·19-s + 4/5·25-s − 0.377·28-s − 1.07·31-s + 0.609·43-s − 2/7·49-s + 0.832·52-s + 1.02·61-s + 7/8·64-s + 0.733·67-s + 0.468·73-s + 0.114·76-s + 1.35·79-s − 1.25·91-s − 1.82·97-s − 2/5·100-s + 2.36·103-s − 1.72·109-s − 0.566·112-s − 1.27·121-s + 0.538·124-s + 0.0887·127-s + 0.0873·131-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  =  $(4,\ 555579,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.244872874$
$L(\frac12)$  $\approx$  $1.244872874$
$L(\frac{3}{2})$   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$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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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

−8.473405990508333938942460162145, −7.979826918997112010936625153086, −7.61815308221120960965862315670, −7.06964187769904693294430669105, −6.80095192435737797106642596261, −6.21210599642288180344551815767, −5.38093432570391435903745350965, −5.20118939724020599935340990965, −4.73935881778659911720376669301, −4.25044865189904014730667776346, −3.74330600536114204681332669925, −2.89600271901229730663785609525, −2.32818830034144713715101440423, −1.76513390485479161742475196655, −0.56983358971280407634685340641, 0.56983358971280407634685340641, 1.76513390485479161742475196655, 2.32818830034144713715101440423, 2.89600271901229730663785609525, 3.74330600536114204681332669925, 4.25044865189904014730667776346, 4.73935881778659911720376669301, 5.20118939724020599935340990965, 5.38093432570391435903745350965, 6.21210599642288180344551815767, 6.80095192435737797106642596261, 7.06964187769904693294430669105, 7.61815308221120960965862315670, 7.979826918997112010936625153086, 8.473405990508333938942460162145

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