Properties

Label 4-640332-1.1-c1e2-0-27
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 − 8·19-s − 2·21-s − 6·25-s − 27-s + 2·28-s − 8·31-s + 36-s − 4·37-s − 4·39-s − 48-s + 3·49-s + 4·52-s + 8·57-s − 28·61-s + 2·63-s + 64-s + 8·67-s + 12·73-s + 6·75-s − 8·76-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 − 1.83·19-s − 0.436·21-s − 6/5·25-s − 0.192·27-s + 0.377·28-s − 1.43·31-s + 1/6·36-s − 0.657·37-s − 0.640·39-s − 0.144·48-s + 3/7·49-s + 0.554·52-s + 1.05·57-s − 3.58·61-s + 0.251·63-s + 1/8·64-s + 0.977·67-s + 1.40·73-s + 0.692·75-s − 0.917·76-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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.036862420200993690310137538450, −7.82491759382859411774440769361, −7.06103937421558916977777144566, −6.91394332771191655575545668718, −6.17569848665337782312463950999, −5.88974476808873252053152463118, −5.62472658216482534683012385296, −4.72832473817225769292057457440, −4.54468764173297491130716278035, −3.72716170584453900058590617219, −3.52268894932194979183855581758, −2.43951823946059783270435316404, −1.87596574858564202101556695192, −1.37340800303348386328024038090, 0, 1.37340800303348386328024038090, 1.87596574858564202101556695192, 2.43951823946059783270435316404, 3.52268894932194979183855581758, 3.72716170584453900058590617219, 4.54468764173297491130716278035, 4.72832473817225769292057457440, 5.62472658216482534683012385296, 5.88974476808873252053152463118, 6.17569848665337782312463950999, 6.91394332771191655575545668718, 7.06103937421558916977777144566, 7.82491759382859411774440769361, 8.036862420200993690310137538450

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