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Properties

Label 4-1045e2-1.1-c0e2-0-1
Degree $4$
Conductor $1092025$
Sign $1$
Analytic cond. $0.271986$
Root an. cond. $0.722165$
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  + 2·5-s − 2·7-s − 16-s + 2·17-s + 2·19-s − 2·23-s + 3·25-s − 4·35-s − 2·43-s + 2·47-s + 2·49-s − 2·73-s − 2·80-s − 81-s + 2·83-s + 4·85-s + 4·95-s + 2·112-s − 4·115-s − 4·119-s − 121-s + 4·125-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + ⋯
L(s)  = 1  + 2·5-s − 2·7-s − 16-s + 2·17-s + 2·19-s − 2·23-s + 3·25-s − 4·35-s − 2·43-s + 2·47-s + 2·49-s − 2·73-s − 2·80-s − 81-s + 2·83-s + 4·85-s + 4·95-s + 2·112-s − 4·115-s − 4·119-s − 121-s + 4·125-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1092025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1092025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1092025\)    =    \(5^{2} \cdot 11^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.271986\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1092025,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.200010447\)
\(L(\frac12)\) \(\approx\) \(1.200010447\)
\(L(1)\) 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)$
bad5$C_1$ \( ( 1 - T )^{2} \)
11$C_2$ \( 1 + T^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
3$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
13$C_2^2$ \( 1 + T^{4} \)
17$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_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
53$C_2^2$ \( 1 + T^{4} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
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

−10.04146358678350719389458911967, −9.750555883041708470138485995596, −9.703819952061487161337903822808, −9.433533239572184542543172121991, −8.763136772773211943346164645880, −8.500907774641200868953012118476, −7.68281396872433663174668837946, −7.25347571617745361085254469402, −7.00609858233756560852249127792, −6.22155327269194202381134855510, −6.15179120064968854027952070936, −5.81256104131291635026301797615, −5.25345105607793365622865073377, −4.98193484147774493703251283530, −3.99803428813241766683371529063, −3.47988804301851918412784710548, −3.00648139115225632376177120955, −2.61898833991937222704504863424, −1.86659777510316916573752527127, −1.12172049518387130788289792091, 1.12172049518387130788289792091, 1.86659777510316916573752527127, 2.61898833991937222704504863424, 3.00648139115225632376177120955, 3.47988804301851918412784710548, 3.99803428813241766683371529063, 4.98193484147774493703251283530, 5.25345105607793365622865073377, 5.81256104131291635026301797615, 6.15179120064968854027952070936, 6.22155327269194202381134855510, 7.00609858233756560852249127792, 7.25347571617745361085254469402, 7.68281396872433663174668837946, 8.500907774641200868953012118476, 8.763136772773211943346164645880, 9.433533239572184542543172121991, 9.703819952061487161337903822808, 9.750555883041708470138485995596, 10.04146358678350719389458911967

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