Properties

Degree $4$
Conductor $2052$
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 − 4·7-s + 9-s − 12-s − 2·13-s + 16-s − 19-s − 4·21-s + 6·25-s + 27-s + 4·28-s − 6·31-s − 36-s + 6·37-s − 2·39-s + 8·43-s + 48-s − 2·49-s + 2·52-s − 57-s + 4·61-s − 4·63-s − 64-s − 4·67-s + 8·73-s + 6·75-s + ⋯
L(s)  = 1  + 0.577·3-s − 1/2·4-s − 1.51·7-s + 1/3·9-s − 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.229·19-s − 0.872·21-s + 6/5·25-s + 0.192·27-s + 0.755·28-s − 1.07·31-s − 1/6·36-s + 0.986·37-s − 0.320·39-s + 1.21·43-s + 0.144·48-s − 2/7·49-s + 0.277·52-s − 0.132·57-s + 0.512·61-s − 0.503·63-s − 1/8·64-s − 0.488·67-s + 0.936·73-s + 0.692·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2052 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2052 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2052\)    =    \(2^{2} \cdot 3^{3} \cdot 19\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{2052} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2052,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6330336146\)
\(L(\frac12)\) \(\approx\) \(0.6330336146\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 + T^{2} \)
3$C_1$ \( 1 - T \)
19$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−13.12982016308530389779564687836, −12.75257719226727857302416322486, −12.47883099828502725741267931384, −11.44396692526093626961716135099, −10.65321118649369174523714247393, −9.990951051761340296038795114574, −9.412638772694111810712427326455, −9.035608504117036944858830230514, −8.184581718073056415429808612585, −7.33686756601273619640602659882, −6.66774689303465791186697127725, −5.81759490658889932618405137187, −4.72065678272510358447082862123, −3.69714913649530985757996284009, −2.75661394944112353474896301836, 2.75661394944112353474896301836, 3.69714913649530985757996284009, 4.72065678272510358447082862123, 5.81759490658889932618405137187, 6.66774689303465791186697127725, 7.33686756601273619640602659882, 8.184581718073056415429808612585, 9.035608504117036944858830230514, 9.412638772694111810712427326455, 9.990951051761340296038795114574, 10.65321118649369174523714247393, 11.44396692526093626961716135099, 12.47883099828502725741267931384, 12.75257719226727857302416322486, 13.12982016308530389779564687836

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