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Properties

Label 4-3072e2-1.1-c0e2-0-3
Degree $4$
Conductor $9437184$
Sign $1$
Analytic cond. $2.35048$
Root an. cond. $1.23819$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 9-s + 2·11-s + 2·19-s + 2·43-s + 2·49-s − 2·59-s + 2·67-s + 81-s + 2·83-s − 4·89-s − 2·99-s − 2·107-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·171-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 9-s + 2·11-s + 2·19-s + 2·43-s + 2·49-s − 2·59-s + 2·67-s + 81-s + 2·83-s − 4·89-s − 2·99-s − 2·107-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·171-s + 173-s + 179-s + 181-s + 191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9437184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9437184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9437184\)    =    \(2^{20} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(2.35048\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9437184,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.657324162\)
\(L(\frac12)\) \(\approx\) \(1.657324162\)
\(L(1)\) 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)$
bad2 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
good5$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2^2$ \( 1 + T^{4} \)
59$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
89$C_1$ \( ( 1 + T )^{4} \)
97$C_2$ \( ( 1 + T^{2} )^{2} \)
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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

−9.120861765805167954778907475914, −8.742067011689231100751771517544, −8.480720198105445650540411152298, −7.77401081529444014647453021829, −7.69609808001593294661404565500, −7.23648074327990459567691904638, −6.69901979833410237389365337597, −6.56792198876660692623188299498, −5.91338130275833326755794402507, −5.78976522458216297396938701614, −5.23607029821142325538584616136, −5.01203155720588031419351183405, −4.16425258856899897415220870293, −4.00970815369222455794927578997, −3.64573613087210841047816004284, −2.92104513493754761761269315478, −2.78031566266009699371606239922, −2.04257829060276103367834255572, −1.27070070165562505454868999107, −0.943485650547084277800167327003, 0.943485650547084277800167327003, 1.27070070165562505454868999107, 2.04257829060276103367834255572, 2.78031566266009699371606239922, 2.92104513493754761761269315478, 3.64573613087210841047816004284, 4.00970815369222455794927578997, 4.16425258856899897415220870293, 5.01203155720588031419351183405, 5.23607029821142325538584616136, 5.78976522458216297396938701614, 5.91338130275833326755794402507, 6.56792198876660692623188299498, 6.69901979833410237389365337597, 7.23648074327990459567691904638, 7.69609808001593294661404565500, 7.77401081529444014647453021829, 8.480720198105445650540411152298, 8.742067011689231100751771517544, 9.120861765805167954778907475914

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