Properties

Degree $4$
Conductor $1494108$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 7-s + 9-s − 4·11-s + 12·13-s + 16-s + 4·17-s − 8·19-s + 16·23-s − 6·25-s − 28-s + 36-s − 20·37-s − 12·41-s − 4·44-s + 49-s + 12·52-s + 12·53-s + 12·61-s − 63-s + 64-s + 8·67-s + 4·68-s + 16·71-s + 20·73-s − 8·76-s + 4·77-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 3.32·13-s + 1/4·16-s + 0.970·17-s − 1.83·19-s + 3.33·23-s − 6/5·25-s − 0.188·28-s + 1/6·36-s − 3.28·37-s − 1.87·41-s − 0.603·44-s + 1/7·49-s + 1.66·52-s + 1.64·53-s + 1.53·61-s − 0.125·63-s + 1/8·64-s + 0.977·67-s + 0.485·68-s + 1.89·71-s + 2.34·73-s − 0.917·76-s + 0.455·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1494108\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{3} \cdot 11^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1494108} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1494108,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.752999213\)
\(L(\frac12)\) \(\approx\) \(2.752999213\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( 1 + T \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−8.209722972899758923826863695247, −7.36329223436070087794310634517, −6.84552480602986516155667793601, −6.61396756737736082376174832588, −6.46232054545852935313207935977, −5.53755315525818574410350601733, −5.33985014787602837985094112170, −5.14287919199214033627107465191, −4.05522826801866843177122831386, −3.62482887081886485478101246558, −3.49945591428017159980646979643, −2.83312116915105400792054837352, −2.03686460629757130002584503967, −1.48110886543319454700611992720, −0.75184239042057648945425095713, 0.75184239042057648945425095713, 1.48110886543319454700611992720, 2.03686460629757130002584503967, 2.83312116915105400792054837352, 3.49945591428017159980646979643, 3.62482887081886485478101246558, 4.05522826801866843177122831386, 5.14287919199214033627107465191, 5.33985014787602837985094112170, 5.53755315525818574410350601733, 6.46232054545852935313207935977, 6.61396756737736082376174832588, 6.84552480602986516155667793601, 7.36329223436070087794310634517, 8.209722972899758923826863695247

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