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Properties

Label 4-1332e2-1.1-c0e2-0-2
Degree $4$
Conductor $1774224$
Sign $1$
Analytic cond. $0.441898$
Root an. cond. $0.815324$
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  − 2·7-s + 2·17-s + 2·19-s + 2·23-s + 2·29-s − 2·47-s + 49-s − 2·53-s − 2·71-s − 2·79-s + 2·83-s + 2·89-s − 2·109-s − 2·113-s − 4·119-s + 121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4·161-s + 163-s + 167-s + ⋯
L(s)  = 1  − 2·7-s + 2·17-s + 2·19-s + 2·23-s + 2·29-s − 2·47-s + 49-s − 2·53-s − 2·71-s − 2·79-s + 2·83-s + 2·89-s − 2·109-s − 2·113-s − 4·119-s + 121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4·161-s + 163-s + 167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1774224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1774224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1774224\)    =    \(2^{4} \cdot 3^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(0.441898\)
Root analytic conductor: \(0.815324\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1774224,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9803440403\)
\(L(\frac12)\) \(\approx\) \(0.9803440403\)
\(L(1)\) 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 \( 1 \)
37$C_2$ \( 1 + T^{2} \)
good5$C_2^2$ \( 1 + T^{4} \)
7$C_2$ \( ( 1 + T + T^{2} )^{2} \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
29$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
31$C_2^2$ \( 1 + T^{4} \)
41$C_2^2$ \( 1 - T^{2} + T^{4} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2$ \( ( 1 + T + T^{2} )^{2} \)
53$C_2$ \( ( 1 + T + T^{2} )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_2$ \( ( 1 + T + T^{2} )^{2} \)
73$C_2^2$ \( 1 - T^{2} + T^{4} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
83$C_2$ \( ( 1 - T + T^{2} )^{2} \)
89$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−10.03004208454528131236286528271, −9.495176796307526365935178089189, −9.298426370763848530811450167283, −9.170113604138990475444606838205, −8.184676394905054721283816671312, −8.133082234759844983683208198924, −7.51902281514363926531244768445, −7.15344682254825108313963550289, −6.57289365047428225011075757892, −6.54872252403708265350922844956, −5.86726542001514331009570654582, −5.51871430423461080657967806116, −4.85476460581243332753220651953, −4.78343392773693847815040386442, −3.74602960213799912227260790124, −3.17552760762104351355673672730, −3.04043126554928920221463277642, −2.94629219843390123621995246818, −1.49503390060530460768852581895, −0.948801273017055576328193321162, 0.948801273017055576328193321162, 1.49503390060530460768852581895, 2.94629219843390123621995246818, 3.04043126554928920221463277642, 3.17552760762104351355673672730, 3.74602960213799912227260790124, 4.78343392773693847815040386442, 4.85476460581243332753220651953, 5.51871430423461080657967806116, 5.86726542001514331009570654582, 6.54872252403708265350922844956, 6.57289365047428225011075757892, 7.15344682254825108313963550289, 7.51902281514363926531244768445, 8.133082234759844983683208198924, 8.184676394905054721283816671312, 9.170113604138990475444606838205, 9.298426370763848530811450167283, 9.495176796307526365935178089189, 10.03004208454528131236286528271

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