Properties

Degree 4
Conductor $ 23^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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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  :  induced by $\chi_{23} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 529,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.248431$
$L(\frac12)$  $\approx$  $0.248431$
$L(\frac{3}{2})$   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$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
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$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
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$C_2^2$ \( 1 + 89 T^{2} + p^{2} T^{4} \)
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

−18.47750850800061646009902647786, −17.90710782011974663086496588020, −17.08205862019940285982025794172, −16.63291171064781582535205855744, −15.65538270528692681049757116794, −15.46280104465881945233209269948, −14.40066184005226474772481053609, −13.96241428320228018659273252765, −12.90390695537994036129091050107, −12.89016809230386862701948159604, −11.46302391247974405152532879935, −11.06445498538988679371642241803, −10.29026506954357994195204927063, −9.462792302698072156583043638081, −8.321948106011545948491366974293, −8.278190574714781685082992842254, −7.56305052056930034956656701516, −5.81727025604144459401902615206, −4.87645242907439515159200857913, −3.67521748643984848866430372052, 3.67521748643984848866430372052, 4.87645242907439515159200857913, 5.81727025604144459401902615206, 7.56305052056930034956656701516, 8.278190574714781685082992842254, 8.321948106011545948491366974293, 9.462792302698072156583043638081, 10.29026506954357994195204927063, 11.06445498538988679371642241803, 11.46302391247974405152532879935, 12.89016809230386862701948159604, 12.90390695537994036129091050107, 13.96241428320228018659273252765, 14.40066184005226474772481053609, 15.46280104465881945233209269948, 15.65538270528692681049757116794, 16.63291171064781582535205855744, 17.08205862019940285982025794172, 17.90710782011974663086496588020, 18.47750850800061646009902647786

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