Properties

Label 4-1050-1.1-c1e2-0-0
Degree $4$
Conductor $1050$
Sign $1$
Analytic cond. $0.0669489$
Root an. cond. $0.508669$
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 − 4-s − 7-s + 2·8-s − 2·9-s + 12-s − 8·13-s + 16-s + 4·17-s + 4·19-s + 21-s − 2·24-s + 25-s + 2·27-s + 28-s + 4·29-s + 8·31-s − 4·32-s + 2·36-s − 20·37-s + 8·39-s + 12·41-s + 8·43-s + 16·47-s − 48-s − 6·49-s − 4·51-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s − 0.377·7-s + 0.707·8-s − 2/3·9-s + 0.288·12-s − 2.21·13-s + 1/4·16-s + 0.970·17-s + 0.917·19-s + 0.218·21-s − 0.408·24-s + 1/5·25-s + 0.384·27-s + 0.188·28-s + 0.742·29-s + 1.43·31-s − 0.707·32-s + 1/3·36-s − 3.28·37-s + 1.28·39-s + 1.87·41-s + 1.21·43-s + 2.33·47-s − 0.144·48-s − 6/7·49-s − 0.560·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.0669489\)
Root analytic conductor: \(0.508669\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1050,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4132844646\)
\(L(\frac12)\) \(\approx\) \(0.4132844646\)
\(L(\frac{3}{2})\) 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_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−19.5318859919, −19.3881993684, −19.1104624590, −18.1058537571, −17.4014938000, −17.2359534775, −16.8198467621, −15.9590260302, −15.5600040575, −14.5358664373, −14.0069258505, −13.8941777274, −12.6461787614, −12.1470736931, −11.9657271713, −10.6789224512, −10.3759943109, −9.52345167581, −8.97764609237, −7.66488013442, −7.45420248730, −6.17616144092, −5.23920392625, −4.63923341950, −3.00381372509, 3.00381372509, 4.63923341950, 5.23920392625, 6.17616144092, 7.45420248730, 7.66488013442, 8.97764609237, 9.52345167581, 10.3759943109, 10.6789224512, 11.9657271713, 12.1470736931, 12.6461787614, 13.8941777274, 14.0069258505, 14.5358664373, 15.5600040575, 15.9590260302, 16.8198467621, 17.2359534775, 17.4014938000, 18.1058537571, 19.1104624590, 19.3881993684, 19.5318859919

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