Properties

Label 4-270000-1.1-c1e2-0-4
Degree $4$
Conductor $270000$
Sign $1$
Analytic cond. $17.2154$
Root an. cond. $2.03694$
Motivic weight $1$
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 − 3-s + 2·4-s − 2·6-s + 9-s + 4·11-s − 2·12-s + 2·13-s − 4·16-s + 2·18-s + 8·22-s + 12·23-s + 4·26-s − 27-s − 8·32-s − 4·33-s + 2·36-s + 4·37-s − 2·39-s + 8·44-s + 24·46-s + 4·47-s + 4·48-s − 5·49-s + 4·52-s − 2·54-s − 20·59-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s + 1.20·11-s − 0.577·12-s + 0.554·13-s − 16-s + 0.471·18-s + 1.70·22-s + 2.50·23-s + 0.784·26-s − 0.192·27-s − 1.41·32-s − 0.696·33-s + 1/3·36-s + 0.657·37-s − 0.320·39-s + 1.20·44-s + 3.53·46-s + 0.583·47-s + 0.577·48-s − 5/7·49-s + 0.554·52-s − 0.272·54-s − 2.60·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(270000\)    =    \(2^{4} \cdot 3^{3} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(17.2154\)
Root analytic conductor: \(2.03694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 270000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.407148812\)
\(L(\frac12)\) \(\approx\) \(3.407148812\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
3$C_1$ \( 1 + T \)
5 \( 1 \)
good7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \)
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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

−8.979440455637625961269202062502, −8.686698967307485893609350306598, −7.62852485359829009908403777253, −7.32922657340821485296829402521, −6.79375443883521821659151920822, −6.27130274152454875896945514364, −6.05569215012514529764032801313, −5.48558822518018321489922753080, −4.75727383162943095433406736583, −4.62309319523854591570832526017, −3.98830812112494448268150912948, −3.24466149791278329969031792942, −3.01327782908855925530719016170, −1.87603850679094242426395746447, −0.985333224458569944413443756671, 0.985333224458569944413443756671, 1.87603850679094242426395746447, 3.01327782908855925530719016170, 3.24466149791278329969031792942, 3.98830812112494448268150912948, 4.62309319523854591570832526017, 4.75727383162943095433406736583, 5.48558822518018321489922753080, 6.05569215012514529764032801313, 6.27130274152454875896945514364, 6.79375443883521821659151920822, 7.32922657340821485296829402521, 7.62852485359829009908403777253, 8.686698967307485893609350306598, 8.979440455637625961269202062502

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