Properties

Degree 4
Conductor $ 2^{7} \cdot 3^{2} \cdot 19^{2} $
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 − 3-s + 4-s − 8·5-s − 6-s + 8-s − 2·9-s − 8·10-s − 12-s + 8·15-s + 16-s − 2·18-s − 2·19-s − 8·20-s − 2·23-s − 24-s + 38·25-s + 5·27-s − 10·29-s + 8·30-s + 32-s − 2·36-s − 2·38-s − 8·40-s + 8·43-s + 16·45-s − 2·46-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 3.57·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s − 2.52·10-s − 0.288·12-s + 2.06·15-s + 1/4·16-s − 0.471·18-s − 0.458·19-s − 1.78·20-s − 0.417·23-s − 0.204·24-s + 38/5·25-s + 0.962·27-s − 1.85·29-s + 1.46·30-s + 0.176·32-s − 1/3·36-s − 0.324·38-s − 1.26·40-s + 1.21·43-s + 2.38·45-s − 0.294·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(415872\)    =    \(2^{7} \cdot 3^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{415872} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((4,\ 415872,\ (\ :1/2, 1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;19\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( 1 - T \)
3$C_2$ \( 1 + T + p T^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−8.408486909011940238981299566497, −7.69091964872191681066145092749, −7.51365445718420633059750043792, −7.25038064772684706053554512263, −6.52592917100039061776544750020, −6.14974687185144050865472542876, −5.28069074518443602737935578596, −5.05050378173441072659755683035, −4.30490757572779981002207201168, −3.86057015395193111701613139713, −3.81281010152338816670306094022, −3.08766645378616687637023050824, −2.40005554242355398519906349614, −0.807536518397022830754994430552, 0, 0.807536518397022830754994430552, 2.40005554242355398519906349614, 3.08766645378616687637023050824, 3.81281010152338816670306094022, 3.86057015395193111701613139713, 4.30490757572779981002207201168, 5.05050378173441072659755683035, 5.28069074518443602737935578596, 6.14974687185144050865472542876, 6.52592917100039061776544750020, 7.25038064772684706053554512263, 7.51365445718420633059750043792, 7.69091964872191681066145092749, 8.408486909011940238981299566497

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