Properties

Degree 4
Conductor $ 2^{7} \cdot 13^{3} $
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 + 4-s − 2·5-s + 8-s + 3·9-s − 2·10-s − 4·11-s − 13-s + 16-s − 6·17-s + 3·18-s + 12·19-s − 2·20-s − 4·22-s − 8·23-s − 7·25-s − 26-s + 32-s − 6·34-s + 3·36-s + 6·37-s + 12·38-s − 2·40-s − 4·44-s − 6·45-s − 8·46-s − 13·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s + 9-s − 0.632·10-s − 1.20·11-s − 0.277·13-s + 1/4·16-s − 1.45·17-s + 0.707·18-s + 2.75·19-s − 0.447·20-s − 0.852·22-s − 1.66·23-s − 7/5·25-s − 0.196·26-s + 0.176·32-s − 1.02·34-s + 1/2·36-s + 0.986·37-s + 1.94·38-s − 0.316·40-s − 0.603·44-s − 0.894·45-s − 1.17·46-s − 1.85·49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 281216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 281216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(281216\)    =    \(2^{7} \cdot 13^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{281216} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 281216,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

Imaginary part of the first few zeros on the critical line

−8.372504302205780920168328513074, −7.952773675345803042836433200054, −7.54927518755809077497495838664, −7.45985560667588965978165906144, −6.83793489243240202981608052708, −5.99269128295739404934949048165, −5.88908450915949561478136129218, −5.03252986183832812949281939535, −4.61153453491182141362175201622, −4.25770102664182839693622055004, −3.54070244821137046528273748949, −3.09063283442029069651336655846, −2.27841740887741805213486299325, −1.52941733184663708479315761299, 0, 1.52941733184663708479315761299, 2.27841740887741805213486299325, 3.09063283442029069651336655846, 3.54070244821137046528273748949, 4.25770102664182839693622055004, 4.61153453491182141362175201622, 5.03252986183832812949281939535, 5.88908450915949561478136129218, 5.99269128295739404934949048165, 6.83793489243240202981608052708, 7.45985560667588965978165906144, 7.54927518755809077497495838664, 7.952773675345803042836433200054, 8.372504302205780920168328513074

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