Properties

Label 4-640332-1.1-c1e2-0-21
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 + 4·13-s + 16-s − 16·19-s + 2·21-s − 6·25-s − 27-s − 2·28-s + 16·31-s + 36-s + 12·37-s − 4·39-s + 16·43-s − 48-s + 3·49-s + 4·52-s + 16·57-s − 28·61-s − 2·63-s + 64-s − 8·67-s − 28·73-s + 6·75-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s − 0.755·7-s + 1/3·9-s − 0.288·12-s + 1.10·13-s + 1/4·16-s − 3.67·19-s + 0.436·21-s − 6/5·25-s − 0.192·27-s − 0.377·28-s + 2.87·31-s + 1/6·36-s + 1.97·37-s − 0.640·39-s + 2.43·43-s − 0.144·48-s + 3/7·49-s + 0.554·52-s + 2.11·57-s − 3.58·61-s − 0.251·63-s + 1/8·64-s − 0.977·67-s − 3.27·73-s + 0.692·75-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: $\chi_{640332} (1, \cdot )$
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$C_1$ \( ( 1 + T )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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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

−8.260375317206775288322444616854, −7.54842093789349707312948138248, −7.34608818311813371377464448078, −6.41309505638029156456085338806, −6.21868608096758870562967590194, −6.10611924951092571905001273028, −5.83815977597462037088554178569, −4.52496529634501691830241759811, −4.27546340147020629083452713682, −4.24578358603152871833997851063, −3.11903064072712609997880205813, −2.65895736335338182549695498478, −2.02661744514455988507183410743, −1.14483701940607571217981775775, 0, 1.14483701940607571217981775775, 2.02661744514455988507183410743, 2.65895736335338182549695498478, 3.11903064072712609997880205813, 4.24578358603152871833997851063, 4.27546340147020629083452713682, 4.52496529634501691830241759811, 5.83815977597462037088554178569, 6.10611924951092571905001273028, 6.21868608096758870562967590194, 6.41309505638029156456085338806, 7.34608818311813371377464448078, 7.54842093789349707312948138248, 8.260375317206775288322444616854

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