Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} \cdot 7^{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  + 3-s − 4·4-s + 2·7-s − 2·9-s − 4·12-s + 10·13-s + 12·16-s + 4·19-s + 2·21-s + 25-s − 5·27-s − 8·28-s − 8·31-s + 8·36-s + 4·37-s + 10·39-s − 20·43-s + 12·48-s + 3·49-s − 40·52-s + 4·57-s + 16·61-s − 4·63-s − 32·64-s − 8·67-s + 4·73-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 2·4-s + 0.755·7-s − 2/3·9-s − 1.15·12-s + 2.77·13-s + 3·16-s + 0.917·19-s + 0.436·21-s + 1/5·25-s − 0.962·27-s − 1.51·28-s − 1.43·31-s + 4/3·36-s + 0.657·37-s + 1.60·39-s − 3.04·43-s + 1.73·48-s + 3/7·49-s − 5.54·52-s + 0.529·57-s + 2.04·61-s − 0.503·63-s − 4·64-s − 0.977·67-s + 0.468·73-s + 0.115·75-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(11025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{11025} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 11025,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.8948642832$
$L(\frac12)$  $\approx$  $0.8948642832$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_2$ \( 1 - T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−11.50769791186352205274303435148, −10.81642134200306905635056067311, −10.26175613408033680908306725927, −9.502060864672765915422726581101, −9.001872818363131182920485953111, −8.679498853252053444462430241617, −8.087023964728641191414598204496, −7.980947296320422939728903060714, −6.69771158952637091109528432278, −5.63914839037632160871019437372, −5.47907564040983295563221596641, −4.54321060660836499326534703030, −3.61689003045514298236484607491, −3.45067056060882590557565434961, −1.35738749247200780270220785705, 1.35738749247200780270220785705, 3.45067056060882590557565434961, 3.61689003045514298236484607491, 4.54321060660836499326534703030, 5.47907564040983295563221596641, 5.63914839037632160871019437372, 6.69771158952637091109528432278, 7.980947296320422939728903060714, 8.087023964728641191414598204496, 8.679498853252053444462430241617, 9.001872818363131182920485953111, 9.502060864672765915422726581101, 10.26175613408033680908306725927, 10.81642134200306905635056067311, 11.50769791186352205274303435148

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