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Properties

Label 4-1323-1.1-c1e2-0-0
Degree $4$
Conductor $1323$
Sign $1$
Analytic cond. $0.0843556$
Root an. cond. $0.538925$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·4-s − 2·7-s + 9-s − 3·12-s − 4·13-s + 5·16-s + 8·19-s − 2·21-s − 6·25-s + 27-s + 6·28-s − 3·36-s + 12·37-s − 4·39-s − 8·43-s + 5·48-s + 3·49-s + 12·52-s + 8·57-s − 4·61-s − 2·63-s − 3·64-s + 8·67-s − 12·73-s − 6·75-s − 24·76-s + ⋯
L(s)  = 1  + 0.577·3-s − 3/2·4-s − 0.755·7-s + 1/3·9-s − 0.866·12-s − 1.10·13-s + 5/4·16-s + 1.83·19-s − 0.436·21-s − 6/5·25-s + 0.192·27-s + 1.13·28-s − 1/2·36-s + 1.97·37-s − 0.640·39-s − 1.21·43-s + 0.721·48-s + 3/7·49-s + 1.66·52-s + 1.05·57-s − 0.512·61-s − 0.251·63-s − 3/8·64-s + 0.977·67-s − 1.40·73-s − 0.692·75-s − 2.75·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.0843556\)
Root analytic conductor: \(0.538925\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1323,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4977203473\)
\(L(\frac12)\) \(\approx\) \(0.4977203473\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
7$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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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

−13.77909280462587463511606590839, −13.19379250527463570089741680145, −13.00543957503168228446980436667, −12.02428486441375050992152755294, −11.55595081754559805664353277943, −10.18331996404873608095625884352, −9.724661210726173973340366981146, −9.457705101580370517739336150901, −8.672365196683144779961098572501, −7.81295430367747747098376616892, −7.25047783802838427330028450541, −5.94607665445287604157300008429, −5.02709416296405066539370067618, −4.13559084050773741974089362056, −3.05422074105458389777226041971, 3.05422074105458389777226041971, 4.13559084050773741974089362056, 5.02709416296405066539370067618, 5.94607665445287604157300008429, 7.25047783802838427330028450541, 7.81295430367747747098376616892, 8.672365196683144779961098572501, 9.457705101580370517739336150901, 9.724661210726173973340366981146, 10.18331996404873608095625884352, 11.55595081754559805664353277943, 12.02428486441375050992152755294, 13.00543957503168228446980436667, 13.19379250527463570089741680145, 13.77909280462587463511606590839

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