Properties

Label 4-567e2-1.1-c0e2-0-6
Degree $4$
Conductor $321489$
Sign $1$
Analytic cond. $0.0800719$
Root an. cond. $0.531949$
Motivic weight $0$
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  + 4-s + 7-s − 25-s + 28-s − 4·37-s + 2·43-s − 64-s − 2·67-s − 2·79-s − 100-s − 4·109-s + 121-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 2·172-s + 173-s − 175-s + 179-s + ⋯
L(s)  = 1  + 4-s + 7-s − 25-s + 28-s − 4·37-s + 2·43-s − 64-s − 2·67-s − 2·79-s − 100-s − 4·109-s + 121-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 2·172-s + 173-s − 175-s + 179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(321489\)    =    \(3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.0800719\)
Root analytic conductor: \(0.531949\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 321489,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.057627678\)
\(L(\frac12)\) \(\approx\) \(1.057627678\)
\(L(1)\) 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)$
bad3 \( 1 \)
7$C_2$ \( 1 - T + T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + T^{4} \)
5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_2^2$ \( 1 - T^{2} + T^{4} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_1$ \( ( 1 + T )^{4} \)
41$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
43$C_2$ \( ( 1 - T + T^{2} )^{2} \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 + T + T^{2} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2$ \( ( 1 + T + T^{2} )^{2} \)
83$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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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

−10.99940424432874667142901200152, −10.90247207123009149946381141655, −10.37637298825697509188547777643, −10.11069840414037343952479266800, −9.215880219020450968942326803420, −9.128346622071799831094325365762, −8.334820930453629618858090074600, −8.222153128999321471228880011665, −7.40060848009186530620805562733, −7.25944728115866275533600091715, −6.80111424525177415749482258036, −6.20764933553850848495352885464, −5.50263309187791770556900334756, −5.44727404574104740297571047752, −4.50423450333269919520417536148, −4.19673234047705220604607422569, −3.34247075571284060273309278450, −2.80538349100618224694946826426, −1.86503485769790619053707558087, −1.67920132820486876628609592671, 1.67920132820486876628609592671, 1.86503485769790619053707558087, 2.80538349100618224694946826426, 3.34247075571284060273309278450, 4.19673234047705220604607422569, 4.50423450333269919520417536148, 5.44727404574104740297571047752, 5.50263309187791770556900334756, 6.20764933553850848495352885464, 6.80111424525177415749482258036, 7.25944728115866275533600091715, 7.40060848009186530620805562733, 8.222153128999321471228880011665, 8.334820930453629618858090074600, 9.128346622071799831094325365762, 9.215880219020450968942326803420, 10.11069840414037343952479266800, 10.37637298825697509188547777643, 10.90247207123009149946381141655, 10.99940424432874667142901200152

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