Properties

Label 4-1440e2-1.1-c1e2-0-20
Degree $4$
Conductor $2073600$
Sign $1$
Analytic cond. $132.214$
Root an. cond. $3.39093$
Motivic weight $1$
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  + 4·5-s − 4·7-s − 2·13-s + 10·17-s − 8·19-s + 4·23-s + 11·25-s − 16·35-s + 2·37-s + 12·43-s − 4·47-s + 8·49-s + 14·53-s + 8·59-s − 8·61-s − 8·65-s + 20·67-s − 6·73-s − 32·79-s − 4·83-s + 40·85-s + 8·91-s − 32·95-s − 6·97-s + 12·101-s + 12·103-s + 12·107-s + ⋯
L(s)  = 1  + 1.78·5-s − 1.51·7-s − 0.554·13-s + 2.42·17-s − 1.83·19-s + 0.834·23-s + 11/5·25-s − 2.70·35-s + 0.328·37-s + 1.82·43-s − 0.583·47-s + 8/7·49-s + 1.92·53-s + 1.04·59-s − 1.02·61-s − 0.992·65-s + 2.44·67-s − 0.702·73-s − 3.60·79-s − 0.439·83-s + 4.33·85-s + 0.838·91-s − 3.28·95-s − 0.609·97-s + 1.19·101-s + 1.18·103-s + 1.16·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2073600\)    =    \(2^{10} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(132.214\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2073600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.772366430\)
\(L(\frac12)\) \(\approx\) \(2.772366430\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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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

−9.706368679627671892363660205536, −9.634226765099376950973003763797, −8.866677094136685555256486168361, −8.746030901339129844528319207681, −8.323829790342609236130197318107, −7.53825700585659573809609403726, −7.10765299498488898416790245043, −7.01751896806842307494192525828, −6.18171859291098007877303627515, −6.12225147271920256764736816104, −5.69542037034675962220846327165, −5.39838589141116257519069002845, −4.75726961685288684636032462456, −4.24584519399071809216271442701, −3.52891379735976921078187113415, −3.16919141435582834602372661986, −2.49104297607341325453529706241, −2.32131597177425001208347563223, −1.38383062808974293538513952391, −0.70139675791265232929629813906, 0.70139675791265232929629813906, 1.38383062808974293538513952391, 2.32131597177425001208347563223, 2.49104297607341325453529706241, 3.16919141435582834602372661986, 3.52891379735976921078187113415, 4.24584519399071809216271442701, 4.75726961685288684636032462456, 5.39838589141116257519069002845, 5.69542037034675962220846327165, 6.12225147271920256764736816104, 6.18171859291098007877303627515, 7.01751896806842307494192525828, 7.10765299498488898416790245043, 7.53825700585659573809609403726, 8.323829790342609236130197318107, 8.746030901339129844528319207681, 8.866677094136685555256486168361, 9.634226765099376950973003763797, 9.706368679627671892363660205536

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