Properties

Degree $4$
Conductor $98784$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 8-s + 3·9-s − 2·12-s − 14-s + 16-s + 3·18-s − 8·19-s + 2·21-s − 2·24-s − 6·25-s − 4·27-s − 28-s − 4·29-s + 32-s + 3·36-s − 20·37-s − 8·38-s + 2·42-s − 2·48-s + 49-s − 6·50-s + 12·53-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 9-s − 0.577·12-s − 0.267·14-s + 1/4·16-s + 0.707·18-s − 1.83·19-s + 0.436·21-s − 0.408·24-s − 6/5·25-s − 0.769·27-s − 0.188·28-s − 0.742·29-s + 0.176·32-s + 1/2·36-s − 3.28·37-s − 1.29·38-s + 0.308·42-s − 0.288·48-s + 1/7·49-s − 0.848·50-s + 1.64·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(98784\)    =    \(2^{5} \cdot 3^{2} \cdot 7^{3}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{98784} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 98784,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( 1 - T \)
3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.415781556927177003100714607194, −8.825241684599749435471219697260, −8.356918966632091922574869118017, −7.67454257840429899018734503053, −6.92538722224681151756066962499, −6.84552480602986516155667793601, −6.18874160468740625251832036222, −5.60435712531698215962545180754, −5.33985014787602837985094112170, −4.60067949543777570406002885677, −3.91638973375108536412463827494, −3.62482887081886485478101246558, −2.39883079209175985717649788244, −1.69840313204406173914058397816, 0, 1.69840313204406173914058397816, 2.39883079209175985717649788244, 3.62482887081886485478101246558, 3.91638973375108536412463827494, 4.60067949543777570406002885677, 5.33985014787602837985094112170, 5.60435712531698215962545180754, 6.18874160468740625251832036222, 6.84552480602986516155667793601, 6.92538722224681151756066962499, 7.67454257840429899018734503053, 8.356918966632091922574869118017, 8.825241684599749435471219697260, 9.415781556927177003100714607194

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