Properties

Degree 4
Conductor $ 11^{3} \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 − 4·4-s − 2·5-s − 3·9-s − 11-s + 8·12-s + 4·15-s + 12·16-s + 8·20-s + 14·23-s − 7·25-s + 14·27-s − 6·31-s + 2·33-s + 12·36-s − 22·37-s + 4·44-s + 6·45-s − 8·47-s − 24·48-s − 10·49-s + 4·53-s + 2·55-s − 2·59-s − 16·60-s − 32·64-s − 2·67-s + ⋯
L(s)  = 1  − 1.15·3-s − 2·4-s − 0.894·5-s − 9-s − 0.301·11-s + 2.30·12-s + 1.03·15-s + 3·16-s + 1.78·20-s + 2.91·23-s − 7/5·25-s + 2.69·27-s − 1.07·31-s + 0.348·33-s + 2·36-s − 3.61·37-s + 0.603·44-s + 0.894·45-s − 1.16·47-s − 3.46·48-s − 1.42·49-s + 0.549·53-s + 0.269·55-s − 0.260·59-s − 2.06·60-s − 4·64-s − 0.244·67-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 224939 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 224939 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(224939\)    =    \(11^{3} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{224939} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 224939,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{11,\;13\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{11,\;13\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad11$C_1$ \( 1 + T \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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

−8.585957272776804427928347593039, −8.402895113347107551850017247874, −7.58467071601136985466381130671, −7.21661224338768342466420756206, −6.53335607604279330195808002055, −5.82135849339387357739621546932, −5.34527427677156767829428970271, −5.13861412595458857426626840526, −4.75768041533452306802121891710, −4.00846798854752901415694161809, −3.29304594675951575785139026233, −3.13184491109995397121358058374, −1.37068173806372041641011691213, 0, 0, 1.37068173806372041641011691213, 3.13184491109995397121358058374, 3.29304594675951575785139026233, 4.00846798854752901415694161809, 4.75768041533452306802121891710, 5.13861412595458857426626840526, 5.34527427677156767829428970271, 5.82135849339387357739621546932, 6.53335607604279330195808002055, 7.21661224338768342466420756206, 7.58467071601136985466381130671, 8.402895113347107551850017247874, 8.585957272776804427928347593039

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