Properties

Degree $4$
Conductor $640332$
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  + 2·2-s − 3-s + 3·4-s − 2·6-s + 4·8-s + 9-s − 4·11-s − 3·12-s + 5·16-s + 4·17-s + 2·18-s − 8·22-s − 4·24-s − 6·25-s − 27-s − 4·29-s + 6·32-s + 4·33-s + 8·34-s + 3·36-s − 20·37-s − 12·41-s − 12·44-s − 5·48-s + 49-s − 12·50-s − 4·51-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.816·6-s + 1.41·8-s + 1/3·9-s − 1.20·11-s − 0.866·12-s + 5/4·16-s + 0.970·17-s + 0.471·18-s − 1.70·22-s − 0.816·24-s − 6/5·25-s − 0.192·27-s − 0.742·29-s + 1.06·32-s + 0.696·33-s + 1.37·34-s + 1/2·36-s − 3.28·37-s − 1.87·41-s − 1.80·44-s − 0.721·48-s + 1/7·49-s − 1.69·50-s − 0.560·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(640332\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{640332} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 640332,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( 1 + T \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−8.041794720972076136741968833826, −7.53051297775750802690229426732, −7.00964076662314835861319289378, −6.84552480602986516155667793601, −6.12865280835810962222461264752, −5.62041745043719163028862045607, −5.33985014787602837985094112170, −5.08902516951897907289761559347, −4.49101910497487596621107263337, −3.62482887081886485478101246558, −3.60455949614714316552181761291, −2.84912246773210635007971021129, −2.04610371060201319844060835057, −1.54970928253998777283903070899, 0, 1.54970928253998777283903070899, 2.04610371060201319844060835057, 2.84912246773210635007971021129, 3.60455949614714316552181761291, 3.62482887081886485478101246558, 4.49101910497487596621107263337, 5.08902516951897907289761559347, 5.33985014787602837985094112170, 5.62041745043719163028862045607, 6.12865280835810962222461264752, 6.84552480602986516155667793601, 7.00964076662314835861319289378, 7.53051297775750802690229426732, 8.041794720972076136741968833826

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