Properties

Degree $4$
Conductor $4563$
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-s − 4-s + 9-s − 12-s + 2·13-s − 3·16-s − 10·25-s + 27-s − 36-s + 2·39-s − 8·43-s − 3·48-s + 2·49-s − 2·52-s + 4·61-s + 7·64-s − 10·75-s + 16·79-s + 81-s + 10·100-s + 16·103-s − 108-s + 2·117-s − 10·121-s + 127-s − 8·129-s + 131-s + ⋯
L(s)  = 1  + 0.577·3-s − 1/2·4-s + 1/3·9-s − 0.288·12-s + 0.554·13-s − 3/4·16-s − 2·25-s + 0.192·27-s − 1/6·36-s + 0.320·39-s − 1.21·43-s − 0.433·48-s + 2/7·49-s − 0.277·52-s + 0.512·61-s + 7/8·64-s − 1.15·75-s + 1.80·79-s + 1/9·81-s + 100-s + 1.57·103-s − 0.0962·108-s + 0.184·117-s − 0.909·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4563\)    =    \(3^{3} \cdot 13^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{4563} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4563,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8688739290\)
\(L(\frac12)\) \(\approx\) \(0.8688739290\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( 1 - T \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 154 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.37022875262449636944115326377, −11.75021766055426081014441395158, −11.26214871088196986969621364833, −10.50362814369017161678338427353, −9.822547030359767529196509389130, −9.394110857034161108190630023282, −8.716178163971269186033685676944, −8.187549007456376769469535656463, −7.54986009243878845014659485386, −6.72746551951994728812600570829, −5.98311934439363478319193558532, −5.07052661557595651641559047460, −4.18486743734375640931684483257, −3.47002734719492040287822111515, −2.09536210327137556917765822110, 2.09536210327137556917765822110, 3.47002734719492040287822111515, 4.18486743734375640931684483257, 5.07052661557595651641559047460, 5.98311934439363478319193558532, 6.72746551951994728812600570829, 7.54986009243878845014659485386, 8.187549007456376769469535656463, 8.716178163971269186033685676944, 9.394110857034161108190630023282, 9.822547030359767529196509389130, 10.50362814369017161678338427353, 11.26214871088196986969621364833, 11.75021766055426081014441395158, 12.37022875262449636944115326377

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