Properties

Label 4-270000-1.1-c1e2-0-11
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 $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·7-s + 9-s − 10·13-s + 10·19-s − 2·21-s − 27-s − 2·31-s − 4·37-s + 10·39-s + 2·43-s − 11·49-s − 10·57-s − 26·61-s + 2·63-s − 22·67-s − 4·73-s + 16·79-s + 81-s − 20·91-s + 2·93-s + 14·97-s + 8·103-s − 14·109-s + 4·111-s − 10·117-s + 14·121-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s − 2.77·13-s + 2.29·19-s − 0.436·21-s − 0.192·27-s − 0.359·31-s − 0.657·37-s + 1.60·39-s + 0.304·43-s − 1.57·49-s − 1.32·57-s − 3.32·61-s + 0.251·63-s − 2.68·67-s − 0.468·73-s + 1.80·79-s + 1/9·81-s − 2.09·91-s + 0.207·93-s + 1.42·97-s + 0.788·103-s − 1.34·109-s + 0.379·111-s − 0.924·117-s + 1.27·121-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: $\chi_{270000} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 270000,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
5 \( 1 \)
good7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−8.821114032953874975275414885204, −7.82946935002197249360095495788, −7.65692053420996459201778762115, −7.40049250827422786308554046568, −6.94607178869965744891714265309, −6.17945227400387495932931920930, −5.67959326371779885195862676134, −5.09151237655519270484018689309, −4.74416118025497742788913645945, −4.57579314482224407000052258764, −3.37887843303378899402222918182, −2.97401840613225752286969945588, −2.10805618754928013340727410523, −1.35841755750539912162882790880, 0, 1.35841755750539912162882790880, 2.10805618754928013340727410523, 2.97401840613225752286969945588, 3.37887843303378899402222918182, 4.57579314482224407000052258764, 4.74416118025497742788913645945, 5.09151237655519270484018689309, 5.67959326371779885195862676134, 6.17945227400387495932931920930, 6.94607178869965744891714265309, 7.40049250827422786308554046568, 7.65692053420996459201778762115, 7.82946935002197249360095495788, 8.821114032953874975275414885204

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