Properties

Degree 4
Conductor $ 2^{5} \cdot 13^{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  + 2-s + 4-s − 2·5-s + 8-s + 3·9-s − 2·10-s − 2·13-s + 16-s − 6·17-s + 3·18-s − 2·20-s − 7·25-s − 2·26-s + 4·29-s + 32-s − 6·34-s + 3·36-s + 6·37-s − 2·40-s − 6·45-s − 13·49-s − 7·50-s − 2·52-s + 24·53-s + 4·58-s − 16·61-s + 64-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 − 0.554·13-s + 1/4·16-s − 1.45·17-s + 0.707·18-s − 0.447·20-s − 7/5·25-s − 0.392·26-s + 0.742·29-s + 0.176·32-s − 1.02·34-s + 1/2·36-s + 0.986·37-s − 0.316·40-s − 0.894·45-s − 1.85·49-s − 0.989·50-s − 0.277·52-s + 3.29·53-s + 0.525·58-s − 2.04·61-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(5408\)    =    \(2^{5} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{5408} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 5408,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.120257005$
$L(\frac12)$  $\approx$  $1.120257005$
$L(\frac{3}{2})$   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$C_1$ \( ( 1 + T )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−12.14995362405968847306495338636, −11.49712673239204411601833482255, −11.44138555694141255918714539737, −10.28025762974353066563394220777, −10.16479571855116242915645828857, −9.175534225967753404519516979495, −8.542273196799616534200427297006, −7.53566579558648020530867420842, −7.45985560667588965978165906144, −6.53148501748885385532746785562, −5.88908450915949561478136129218, −4.61153453491182141362175201622, −4.43437896316507372499242030963, −3.47529908355398035375426875224, −2.17545984222807868215828295765, 2.17545984222807868215828295765, 3.47529908355398035375426875224, 4.43437896316507372499242030963, 4.61153453491182141362175201622, 5.88908450915949561478136129218, 6.53148501748885385532746785562, 7.45985560667588965978165906144, 7.53566579558648020530867420842, 8.542273196799616534200427297006, 9.175534225967753404519516979495, 10.16479571855116242915645828857, 10.28025762974353066563394220777, 11.44138555694141255918714539737, 11.49712673239204411601833482255, 12.14995362405968847306495338636

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