Properties

Degree 4
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 2·9-s − 2·11-s + 15-s − 10·23-s − 4·25-s + 2·27-s + 10·31-s + 2·33-s − 10·37-s + 2·45-s − 16·47-s + 8·49-s − 4·53-s + 2·55-s − 4·59-s + 2·67-s + 10·69-s − 16·71-s + 4·75-s + 7·81-s + 22·89-s − 10·93-s − 14·97-s + 4·99-s − 2·103-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.603·11-s + 0.258·15-s − 2.08·23-s − 4/5·25-s + 0.384·27-s + 1.79·31-s + 0.348·33-s − 1.64·37-s + 0.298·45-s − 2.33·47-s + 8/7·49-s − 0.549·53-s + 0.269·55-s − 0.520·59-s + 0.244·67-s + 1.20·69-s − 1.89·71-s + 0.461·75-s + 7/9·81-s + 2.33·89-s − 1.03·93-s − 1.42·97-s + 0.402·99-s − 0.197·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(29040\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{29040} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 29040,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
11$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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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

−10.36374883997169791164214822460, −9.906566539530146269782375770738, −9.297414656799504870432131549516, −8.488460286034259992824918190629, −8.130599524009900109543910465446, −7.75312156167697894020424745322, −6.95986755461883431688224976728, −6.22176796833594975251026998923, −5.91893991817109371697024201968, −5.17178128346823184568413524452, −4.56145089833578904943988733388, −3.77052797663273840924601578362, −2.98711440222433410761523446210, −1.94297767047486301309072781769, 0, 1.94297767047486301309072781769, 2.98711440222433410761523446210, 3.77052797663273840924601578362, 4.56145089833578904943988733388, 5.17178128346823184568413524452, 5.91893991817109371697024201968, 6.22176796833594975251026998923, 6.95986755461883431688224976728, 7.75312156167697894020424745322, 8.130599524009900109543910465446, 8.488460286034259992824918190629, 9.297414656799504870432131549516, 9.906566539530146269782375770738, 10.36374883997169791164214822460

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