Properties

Label 4-1900e2-1.1-c1e2-0-5
Degree $4$
Conductor $3610000$
Sign $1$
Analytic cond. $230.176$
Root an. cond. $3.89507$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·7-s + 2·13-s + 2·19-s + 8·21-s + 2·27-s + 4·31-s − 10·37-s − 4·39-s − 12·41-s − 4·43-s − 12·47-s − 2·49-s + 18·53-s − 4·57-s + 4·61-s − 10·67-s − 12·71-s + 8·73-s − 8·79-s − 81-s − 24·89-s − 8·91-s − 8·93-s + 2·97-s − 12·101-s − 10·103-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.51·7-s + 0.554·13-s + 0.458·19-s + 1.74·21-s + 0.384·27-s + 0.718·31-s − 1.64·37-s − 0.640·39-s − 1.87·41-s − 0.609·43-s − 1.75·47-s − 2/7·49-s + 2.47·53-s − 0.529·57-s + 0.512·61-s − 1.22·67-s − 1.42·71-s + 0.936·73-s − 0.900·79-s − 1/9·81-s − 2.54·89-s − 0.838·91-s − 0.829·93-s + 0.203·97-s − 1.19·101-s − 0.985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3610000\)    =    \(2^{4} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(230.176\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3610000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 2 T + 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 10 T + 96 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 18 T + 184 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 10 T + 156 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 154 T^{2} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 24 T + 310 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 2 T - 48 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−8.855511533935697558017854457238, −8.818686164756380864525594806133, −8.243704919932355411372816219043, −7.938551033620986159704715607338, −7.16762095013087082325732917164, −6.92184041146793511736874997927, −6.45545676362758861993359922643, −6.44578648275118741177808487897, −5.76729417529343455527964321079, −5.39853823930531554967607062417, −5.21542994543241923041358277048, −4.61810810306989325090843943910, −3.89784477540040678313905319587, −3.65016321528989332605373908981, −2.91027176429740671157007530436, −2.89224244568729188193746483028, −1.76397749494652932195956149680, −1.23019904924336113712124714626, 0, 0, 1.23019904924336113712124714626, 1.76397749494652932195956149680, 2.89224244568729188193746483028, 2.91027176429740671157007530436, 3.65016321528989332605373908981, 3.89784477540040678313905319587, 4.61810810306989325090843943910, 5.21542994543241923041358277048, 5.39853823930531554967607062417, 5.76729417529343455527964321079, 6.44578648275118741177808487897, 6.45545676362758861993359922643, 6.92184041146793511736874997927, 7.16762095013087082325732917164, 7.938551033620986159704715607338, 8.243704919932355411372816219043, 8.818686164756380864525594806133, 8.855511533935697558017854457238

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