Properties

Label 4-332928-1.1-c1e2-0-33
Degree $4$
Conductor $332928$
Sign $1$
Analytic cond. $21.2277$
Root an. cond. $2.14647$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s − 4·11-s − 12·13-s − 8·23-s − 10·25-s + 4·27-s + 8·33-s − 8·37-s + 24·39-s + 16·47-s − 14·49-s + 24·61-s + 16·69-s − 24·71-s + 4·73-s + 20·75-s − 11·81-s − 32·83-s − 36·97-s − 4·99-s + 36·107-s + 16·111-s − 12·117-s − 10·121-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 1.20·11-s − 3.32·13-s − 1.66·23-s − 2·25-s + 0.769·27-s + 1.39·33-s − 1.31·37-s + 3.84·39-s + 2.33·47-s − 2·49-s + 3.07·61-s + 1.92·69-s − 2.84·71-s + 0.468·73-s + 2.30·75-s − 1.22·81-s − 3.51·83-s − 3.65·97-s − 0.402·99-s + 3.48·107-s + 1.51·111-s − 1.10·117-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(332928\)    =    \(2^{7} \cdot 3^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(21.2277\)
Root analytic conductor: \(2.14647\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 332928,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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

−8.417457325737154058407963592246, −7.56400176257965263627932886459, −7.36536171038217363811476753812, −7.09070832417543820123167533646, −6.29752496138876049929111994710, −5.74580019409002215172587712055, −5.33745292990969339905963402490, −5.17682892958574806985483154514, −4.39057198741880239625477546712, −4.09383004007619438966415223632, −2.95072257084128076659869010441, −2.43872790837340716861606160980, −1.90305516483713851254985692530, 0, 0, 1.90305516483713851254985692530, 2.43872790837340716861606160980, 2.95072257084128076659869010441, 4.09383004007619438966415223632, 4.39057198741880239625477546712, 5.17682892958574806985483154514, 5.33745292990969339905963402490, 5.74580019409002215172587712055, 6.29752496138876049929111994710, 7.09070832417543820123167533646, 7.36536171038217363811476753812, 7.56400176257965263627932886459, 8.417457325737154058407963592246

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