Properties

Degree 4
Conductor $ 5^{2} \cdot 7^{2} \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s + 11-s − 4·16-s + 8·23-s − 25-s − 7·49-s + 20·53-s + 28·67-s + 16·71-s − 3·99-s + 16·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 12·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 17·169-s + 173-s − 4·176-s + 179-s + 181-s + ⋯
L(s)  = 1  − 9-s + 0.301·11-s − 16-s + 1.66·23-s − 1/5·25-s − 49-s + 2.74·53-s + 3.42·67-s + 1.89·71-s − 0.301·99-s + 1.50·113-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.30·169-s + 0.0760·173-s − 0.301·176-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 148225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 148225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(148225\)    =    \(5^{2} \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{148225} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 148225,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.387801979$
$L(\frac12)$  $\approx$  $1.387801979$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_2$ \( 1 + T^{2} \)
7$C_2$ \( 1 + p T^{2} \)
11$C_2$ \( 1 - T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \)
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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

−9.224414175459041687773963031127, −8.749546641820203741214562547446, −8.493580001103400656477947030685, −7.931979702871659852073370326371, −7.24006600412275574857535482217, −6.79445842090327170160582443731, −6.48006898691393493047746605363, −5.70193838984999107772234534861, −5.23975374025055338202416439938, −4.81112539523175410463567763555, −3.99095407742630478815800076171, −3.46110135681904694119673074968, −2.67138926466593006156481479333, −2.12629538117729852673534058825, −0.800312735555445643372264521122, 0.800312735555445643372264521122, 2.12629538117729852673534058825, 2.67138926466593006156481479333, 3.46110135681904694119673074968, 3.99095407742630478815800076171, 4.81112539523175410463567763555, 5.23975374025055338202416439938, 5.70193838984999107772234534861, 6.48006898691393493047746605363, 6.79445842090327170160582443731, 7.24006600412275574857535482217, 7.931979702871659852073370326371, 8.493580001103400656477947030685, 8.749546641820203741214562547446, 9.224414175459041687773963031127

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