Properties

Label 4-578-1.1-c1e2-0-0
Degree $4$
Conductor $578$
Sign $1$
Analytic cond. $0.0368537$
Root an. cond. $0.438147$
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  − 2·3-s − 4-s − 2·5-s + 2·8-s − 2·9-s + 6·11-s + 2·12-s + 4·15-s + 16-s − 8·19-s + 2·20-s + 4·23-s − 4·24-s − 6·25-s + 10·27-s + 6·29-s − 4·32-s − 12·33-s + 2·36-s − 6·37-s − 4·40-s + 12·43-s − 6·44-s + 4·45-s − 2·48-s + 2·49-s − 12·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s − 0.894·5-s + 0.707·8-s − 2/3·9-s + 1.80·11-s + 0.577·12-s + 1.03·15-s + 1/4·16-s − 1.83·19-s + 0.447·20-s + 0.834·23-s − 0.816·24-s − 6/5·25-s + 1.92·27-s + 1.11·29-s − 0.707·32-s − 2.08·33-s + 1/3·36-s − 0.986·37-s − 0.632·40-s + 1.82·43-s − 0.904·44-s + 0.596·45-s − 0.288·48-s + 2/7·49-s − 1.61·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(578\)    =    \(2 \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(0.0368537\)
Root analytic conductor: \(0.438147\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 578,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2897979046\)
\(L(\frac12)\) \(\approx\) \(0.2897979046\)
\(L(\frac{3}{2})\) 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$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
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$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−19.9621988029, −19.9292301410, −19.1494492613, −19.0177567201, −17.6761697408, −17.4469801976, −16.9473294095, −16.6317948443, −15.7312805996, −15.0693535709, −14.2197849424, −13.9795262067, −12.8595646329, −11.9583441737, −11.9362311444, −10.8489466872, −10.7173409989, −9.28607198826, −8.69568671187, −7.81910395524, −6.65330731261, −5.99911855172, −4.74199315541, −3.90229547123, 3.90229547123, 4.74199315541, 5.99911855172, 6.65330731261, 7.81910395524, 8.69568671187, 9.28607198826, 10.7173409989, 10.8489466872, 11.9362311444, 11.9583441737, 12.8595646329, 13.9795262067, 14.2197849424, 15.0693535709, 15.7312805996, 16.6317948443, 16.9473294095, 17.4469801976, 17.6761697408, 19.0177567201, 19.1494492613, 19.9292301410, 19.9621988029

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