Properties

Degree 4
Conductor $ 23^{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  − 2-s − 2·4-s − 2·5-s + 2·7-s + 3·8-s − 9-s + 2·10-s − 6·11-s + 6·13-s − 2·14-s + 16-s + 6·17-s + 18-s − 4·19-s + 4·20-s + 6·22-s + 2·23-s − 2·25-s − 6·26-s − 4·28-s − 6·29-s − 2·32-s − 6·34-s − 4·35-s + 2·36-s + 2·37-s + 4·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 4-s − 0.894·5-s + 0.755·7-s + 1.06·8-s − 1/3·9-s + 0.632·10-s − 1.80·11-s + 1.66·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.894·20-s + 1.27·22-s + 0.417·23-s − 2/5·25-s − 1.17·26-s − 0.755·28-s − 1.11·29-s − 0.353·32-s − 1.02·34-s − 0.676·35-s + 1/3·36-s + 0.328·37-s + 0.648·38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(529\)    =    \(23^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{529} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 529,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.2484318665$
$L(\frac12)$  $\approx$  $0.2484318665$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 23$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 23$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad23$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
3$V_4$ \( 1 + T^{2} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_4$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$V_4$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T + 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$V_4$ \( 1 + 89 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 10 T + 154 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 20 T + 237 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 22 T + 247 T^{2} - 22 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 22 T + 282 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
89$C_4$ \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 22 T + 270 T^{2} - 22 p T^{3} + p^{2} T^{4} \)
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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

−19.8540099215, −19.1795973161, −18.5138806169, −18.477508508, −17.9071078201, −17.0820586202, −16.6329117106, −15.6553827053, −15.4628010447, −14.4006618401, −13.9624142832, −12.9039069554, −12.8901680923, −11.4630239125, −11.0644549854, −10.2902650695, −9.4627923027, −8.32194810601, −8.27819057471, −7.56305052057, −5.81727025604, −4.87645242907, −3.67521748644, 3.67521748644, 4.87645242907, 5.81727025604, 7.56305052057, 8.27819057471, 8.32194810601, 9.4627923027, 10.2902650695, 11.0644549854, 11.4630239125, 12.8901680923, 12.9039069554, 13.9624142832, 14.4006618401, 15.4628010447, 15.6553827053, 16.6329117106, 17.0820586202, 17.9071078201, 18.477508508, 18.5138806169, 19.1795973161, 19.8540099215

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