Properties

Degree $4$
Conductor $37044$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s − 4·5-s − 7-s + 9-s − 12-s + 4·15-s + 16-s + 4·17-s − 4·20-s + 21-s + 2·25-s − 27-s − 28-s + 4·35-s + 36-s − 20·37-s − 12·41-s − 8·43-s − 4·45-s − 48-s + 49-s − 4·51-s + 8·59-s + 4·60-s − 63-s + 64-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.377·7-s + 1/3·9-s − 0.288·12-s + 1.03·15-s + 1/4·16-s + 0.970·17-s − 0.894·20-s + 0.218·21-s + 2/5·25-s − 0.192·27-s − 0.188·28-s + 0.676·35-s + 1/6·36-s − 3.28·37-s − 1.87·41-s − 1.21·43-s − 0.596·45-s − 0.144·48-s + 1/7·49-s − 0.560·51-s + 1.04·59-s + 0.516·60-s − 0.125·63-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(37044\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{3}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{37044} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 37044,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 + T \)
7$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.24612146790310983021844103291, −9.804201361737724343194425540841, −8.882543700432626086795887126363, −8.356918966632091922574869118017, −7.954464142207805330736993023418, −7.36399840290663473896741338506, −6.84552480602986516155667793601, −6.54855688605176886120203881032, −5.33985014787602837985094112170, −5.32455191890654195761811472253, −4.16943092436514571048096348777, −3.62482887081886485478101246558, −3.17941538941685381741709304651, −1.71238689353652455033144003099, 0, 1.71238689353652455033144003099, 3.17941538941685381741709304651, 3.62482887081886485478101246558, 4.16943092436514571048096348777, 5.32455191890654195761811472253, 5.33985014787602837985094112170, 6.54855688605176886120203881032, 6.84552480602986516155667793601, 7.36399840290663473896741338506, 7.954464142207805330736993023418, 8.356918966632091922574869118017, 8.882543700432626086795887126363, 9.804201361737724343194425540841, 10.24612146790310983021844103291

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