Properties

Label 4-640332-1.1-c1e2-0-28
Degree $4$
Conductor $640332$
Sign $-1$
Analytic cond. $40.8281$
Root an. cond. $2.52778$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 2·7-s + 9-s + 12-s + 16-s − 12·17-s + 2·21-s − 10·25-s + 27-s + 2·28-s + 36-s − 20·37-s + 12·41-s + 16·43-s − 12·47-s + 48-s − 3·49-s − 12·51-s + 2·63-s + 64-s − 8·67-s − 12·68-s − 10·75-s + 28·79-s + 81-s − 24·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s + 0.288·12-s + 1/4·16-s − 2.91·17-s + 0.436·21-s − 2·25-s + 0.192·27-s + 0.377·28-s + 1/6·36-s − 3.28·37-s + 1.87·41-s + 2.43·43-s − 1.75·47-s + 0.144·48-s − 3/7·49-s − 1.68·51-s + 0.251·63-s + 1/8·64-s − 0.977·67-s − 1.45·68-s − 1.15·75-s + 3.15·79-s + 1/9·81-s − 2.63·83-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: $-1$
Analytic conductor: \(40.8281\)
Root analytic conductor: \(2.52778\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 - T \)
7$C_2$ \( 1 - 2 T + p T^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.013104010237496671597514230187, −7.81940691936061331988033870724, −7.29087986975847329500837910780, −6.77506565559004391547494912024, −6.52504829438381665573881626069, −5.85955462295020247251479297795, −5.44687910527563212016596341412, −4.69333471332706230755932945396, −4.36376964393171453387912321319, −3.89761457992935995656155838511, −3.25261158060160886155862976042, −2.37545091357255938865951638025, −2.12020721928430735862314078108, −1.54124369917794574272131224237, 0, 1.54124369917794574272131224237, 2.12020721928430735862314078108, 2.37545091357255938865951638025, 3.25261158060160886155862976042, 3.89761457992935995656155838511, 4.36376964393171453387912321319, 4.69333471332706230755932945396, 5.44687910527563212016596341412, 5.85955462295020247251479297795, 6.52504829438381665573881626069, 6.77506565559004391547494912024, 7.29087986975847329500837910780, 7.81940691936061331988033870724, 8.013104010237496671597514230187

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