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Properties

Label 4-1045e2-1.1-c0e2-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(1.581832255\)
\(L(\frac12)\) \(\approx\) \(1.581832255\)
\(L(1)\) 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

−10.44199131614385292263510131788, −9.928992766231498495283202157915, −9.434508229280321716747687049869, −9.022136469844090951435148453823, −8.677848217376404455181757425027, −8.472694984801156329706877199512, −8.001380704027362578699331707722, −7.31185976698375435565366289992, −6.92250198409690946150153558808, −6.45161028283209600610327061817, −5.91177030800592252780109755831, −5.86651309342698863519271881083, −5.09481923454441098444602380516, −4.68333793336964788747238850670, −4.21119976371123900749858616250, −4.09145438720171200006255364458, −2.49991502080536803072676393644, −2.24463114706879205074679729136, −2.13086677055753496932109963622, −1.38647406043806404452258546928, 1.38647406043806404452258546928, 2.13086677055753496932109963622, 2.24463114706879205074679729136, 2.49991502080536803072676393644, 4.09145438720171200006255364458, 4.21119976371123900749858616250, 4.68333793336964788747238850670, 5.09481923454441098444602380516, 5.86651309342698863519271881083, 5.91177030800592252780109755831, 6.45161028283209600610327061817, 6.92250198409690946150153558808, 7.31185976698375435565366289992, 8.001380704027362578699331707722, 8.472694984801156329706877199512, 8.677848217376404455181757425027, 9.022136469844090951435148453823, 9.434508229280321716747687049869, 9.928992766231498495283202157915, 10.44199131614385292263510131788

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