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Properties

Label 4-1045e2-1.1-c0e2-0-3
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  + 2·2-s + 2·4-s + 2·8-s + 2·11-s − 2·13-s + 3·16-s + 4·22-s − 2·23-s − 25-s − 4·26-s + 4·32-s + 4·44-s − 4·46-s − 2·47-s − 2·50-s − 4·52-s + 4·64-s − 81-s + 4·88-s − 4·92-s − 4·94-s − 2·100-s − 4·104-s − 2·107-s + 3·121-s + 127-s + 4·128-s + ⋯
L(s)  = 1  + 2·2-s + 2·4-s + 2·8-s + 2·11-s − 2·13-s + 3·16-s + 4·22-s − 2·23-s − 25-s − 4·26-s + 4·32-s + 4·44-s − 4·46-s − 2·47-s − 2·50-s − 4·52-s + 4·64-s − 81-s + 4·88-s − 4·92-s − 4·94-s − 2·100-s − 4·104-s − 2·107-s + 3·121-s + 127-s + 4·128-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

\(L(\frac{1}{2})\) \(\approx\) \(3.015344049\)
\(L(\frac12)\) \(\approx\) \(3.015344049\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
19$C_2$ \( 1 + T^{2} \)
good2$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
3$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
13$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
17$C_2^2$ \( 1 + T^{4} \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
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$C_2^2$ \( 1 + T^{4} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
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

−10.18370987162743641083590069238, −9.854345473161023048654013660687, −9.808446244840076116337404848435, −9.292075785195061160221044845461, −8.512757720099854588943270119186, −8.119966507835763453819939323957, −7.68440188662252246586365449730, −7.27972448965396821559649864185, −6.78224605934323013604861424701, −6.40259518971876131588421356605, −5.89629164892603253984312620578, −5.60710096546892564815887618524, −4.98632450209736878733032099082, −4.53478897405041601349273461536, −4.30616758053099226860088634974, −3.75173983637425831397079934493, −3.46975110841508378527812456417, −2.69728799695975153235870854607, −1.98953361268774909305785756183, −1.50623806758096587503904121300, 1.50623806758096587503904121300, 1.98953361268774909305785756183, 2.69728799695975153235870854607, 3.46975110841508378527812456417, 3.75173983637425831397079934493, 4.30616758053099226860088634974, 4.53478897405041601349273461536, 4.98632450209736878733032099082, 5.60710096546892564815887618524, 5.89629164892603253984312620578, 6.40259518971876131588421356605, 6.78224605934323013604861424701, 7.27972448965396821559649864185, 7.68440188662252246586365449730, 8.119966507835763453819939323957, 8.512757720099854588943270119186, 9.292075785195061160221044845461, 9.808446244840076116337404848435, 9.854345473161023048654013660687, 10.18370987162743641083590069238

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