Properties

Degree 4
Conductor $ 229^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4-s + 6·5-s − 3·9-s + 6·11-s − 2·12-s + 12·15-s − 3·16-s − 6·17-s − 2·19-s − 6·20-s + 17·25-s − 14·27-s + 12·33-s + 3·36-s + 4·37-s − 2·43-s − 6·44-s − 18·45-s − 6·48-s + 14·49-s − 12·51-s + 12·53-s + 36·55-s − 4·57-s − 12·60-s + 10·61-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 2.68·5-s − 9-s + 1.80·11-s − 0.577·12-s + 3.09·15-s − 3/4·16-s − 1.45·17-s − 0.458·19-s − 1.34·20-s + 17/5·25-s − 2.69·27-s + 2.08·33-s + 1/2·36-s + 0.657·37-s − 0.304·43-s − 0.904·44-s − 2.68·45-s − 0.866·48-s + 2·49-s − 1.68·51-s + 1.64·53-s + 4.85·55-s − 0.529·57-s − 1.54·60-s + 1.28·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 52441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 52441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(52441\)    =    \(229^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{52441} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 52441,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.549581091$
$L(\frac12)$  $\approx$  $2.549581091$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 229$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 229$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad229$C_2$ \( 1 - 14 T + p T^{2} \)
good2$V_4$ \( 1 + T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$V_4$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
29$V_4$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
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$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$V_4$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$V_4$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
67$V_4$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
73$V_4$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
79$V_4$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−9.933228883600645345117312716392, −9.288995754727251390007447684528, −8.998106452521941953152706182178, −8.768961470526460026351817099329, −8.587851934459013834761684572790, −7.41739122861685143280509277723, −6.71778066430661336888582716183, −6.29205951881109080234144756635, −5.78640405170751959347105161981, −5.42927634164617286025699613855, −4.38492488245376512643072179509, −3.88261708380527167211809567495, −2.66907075111205383917085810778, −2.37938663941908880657930341002, −1.65730355152551961546014863016, 1.65730355152551961546014863016, 2.37938663941908880657930341002, 2.66907075111205383917085810778, 3.88261708380527167211809567495, 4.38492488245376512643072179509, 5.42927634164617286025699613855, 5.78640405170751959347105161981, 6.29205951881109080234144756635, 6.71778066430661336888582716183, 7.41739122861685143280509277723, 8.587851934459013834761684572790, 8.768961470526460026351817099329, 8.998106452521941953152706182178, 9.288995754727251390007447684528, 9.933228883600645345117312716392

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