Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{3} \cdot 19^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 4-s + 7-s + 6·9-s − 4·12-s − 8·13-s + 16-s + 2·19-s − 4·21-s − 10·25-s + 4·27-s + 28-s − 8·31-s + 6·36-s + 32·39-s + 12·41-s + 16·43-s − 4·48-s + 49-s − 8·52-s − 8·57-s − 12·59-s + 6·63-s + 64-s + 40·75-s + 2·76-s − 37·81-s + ⋯
L(s)  = 1  − 2.30·3-s + 1/2·4-s + 0.377·7-s + 2·9-s − 1.15·12-s − 2.21·13-s + 1/4·16-s + 0.458·19-s − 0.872·21-s − 2·25-s + 0.769·27-s + 0.188·28-s − 1.43·31-s + 36-s + 5.12·39-s + 1.87·41-s + 2.43·43-s − 0.577·48-s + 1/7·49-s − 1.10·52-s − 1.05·57-s − 1.56·59-s + 0.755·63-s + 1/8·64-s + 4.61·75-s + 0.229·76-s − 4.11·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495292 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495292 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(495292\)    =    \(2^{2} \cdot 7^{3} \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{495292} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 495292,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.3404027435\)
\(L(\frac12)\)  \(\approx\)  \(0.3404027435\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;19\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( 1 - T \)
19$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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\[\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.455785059232209610704023328839, −7.59398985049814462943070336733, −7.57571100088867902110310233811, −7.22603565182205852620983220987, −6.49846176311500432692401612038, −6.10620473763340062562735487786, −5.57928681742950427486583645839, −5.50672525573589260552367552204, −4.96338806936869690915366473455, −4.37922399648646721676799207374, −3.95792122264195321677075812729, −2.77078716707906865841432089150, −2.42856616031568333915394300355, −1.43866585895250006043461666994, −0.35629273793426275378567663411, 0.35629273793426275378567663411, 1.43866585895250006043461666994, 2.42856616031568333915394300355, 2.77078716707906865841432089150, 3.95792122264195321677075812729, 4.37922399648646721676799207374, 4.96338806936869690915366473455, 5.50672525573589260552367552204, 5.57928681742950427486583645839, 6.10620473763340062562735487786, 6.49846176311500432692401612038, 7.22603565182205852620983220987, 7.57571100088867902110310233811, 7.59398985049814462943070336733, 8.455785059232209610704023328839

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