Properties

Degree 4
Conductor $ 2^{6} \cdot 11^{2} \cdot 13^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·4-s − 3·9-s − 2·11-s + 4·12-s + 4·16-s − 8·17-s + 4·19-s − 9·25-s + 14·27-s + 4·33-s + 6·36-s + 20·41-s − 8·43-s + 4·44-s − 8·48-s − 10·49-s + 16·51-s − 8·57-s − 2·59-s − 8·64-s − 2·67-s + 16·68-s − 32·73-s + 18·75-s − 8·76-s − 4·81-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 9-s − 0.603·11-s + 1.15·12-s + 16-s − 1.94·17-s + 0.917·19-s − 9/5·25-s + 2.69·27-s + 0.696·33-s + 36-s + 3.12·41-s − 1.21·43-s + 0.603·44-s − 1.15·48-s − 1.42·49-s + 2.24·51-s − 1.05·57-s − 0.260·59-s − 64-s − 0.244·67-s + 1.94·68-s − 3.74·73-s + 2.07·75-s − 0.917·76-s − 4/9·81-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1308736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1308736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1308736\)    =    \(2^{6} \cdot 11^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1308736} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 1308736,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;11,\;13\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + p T^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
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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

−7.44648329782426661272373183158, −7.21661224338768342466420756206, −6.46838419268852437057061476829, −5.94370912777399366835769176593, −5.82135849339387357739621546932, −5.49927884962180738090473987706, −4.75768041533452306802121891710, −4.58597385457409318027109585808, −4.10120986415973978790495886343, −3.29304594675951575785139026233, −2.84256580616515370878491454559, −2.21612589981477744409569670299, −1.19900710774742687001457822190, 0, 0, 1.19900710774742687001457822190, 2.21612589981477744409569670299, 2.84256580616515370878491454559, 3.29304594675951575785139026233, 4.10120986415973978790495886343, 4.58597385457409318027109585808, 4.75768041533452306802121891710, 5.49927884962180738090473987706, 5.82135849339387357739621546932, 5.94370912777399366835769176593, 6.46838419268852437057061476829, 7.21661224338768342466420756206, 7.44648329782426661272373183158

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