Properties

Label 4-1620e2-1.1-c1e2-0-3
Degree $4$
Conductor $2624400$
Sign $1$
Analytic cond. $167.334$
Root an. cond. $3.59663$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2·7-s − 2·13-s − 6·17-s + 10·19-s − 3·23-s + 6·29-s − 5·31-s + 2·35-s + 4·37-s − 12·41-s − 8·43-s + 12·47-s + 7·49-s − 6·53-s − 6·59-s + 7·61-s + 2·65-s − 2·67-s + 24·71-s − 32·73-s + 79-s + 15·83-s + 6·85-s − 24·89-s + 4·91-s − 10·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s − 0.554·13-s − 1.45·17-s + 2.29·19-s − 0.625·23-s + 1.11·29-s − 0.898·31-s + 0.338·35-s + 0.657·37-s − 1.87·41-s − 1.21·43-s + 1.75·47-s + 49-s − 0.824·53-s − 0.781·59-s + 0.896·61-s + 0.248·65-s − 0.244·67-s + 2.84·71-s − 3.74·73-s + 0.112·79-s + 1.64·83-s + 0.650·85-s − 2.54·89-s + 0.419·91-s − 1.02·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2624400\)    =    \(2^{4} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(167.334\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2624400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9918678966\)
\(L(\frac12)\) \(\approx\) \(0.9918678966\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
good7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
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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

−9.658046576257505396066782608257, −9.188121844361043554238643821877, −8.894118974858806846509640377690, −8.405355951009289857382128446472, −7.922361300288781335385497222834, −7.67967462989094291475036555323, −7.08543005360257846533748695637, −6.74483944259291106520329136216, −6.66366405066513570604633639399, −5.84411902691293804897692935056, −5.51398481501151096020107827863, −5.11112619081516648838162756916, −4.56164670331893563799858427284, −4.13392715776059607618726701998, −3.63834738003612573374401987901, −3.07280062367362272739357168848, −2.77877919166242687809943407586, −2.06303190498852804834951384172, −1.33428248650444441071186419701, −0.39719415030826636888928434503, 0.39719415030826636888928434503, 1.33428248650444441071186419701, 2.06303190498852804834951384172, 2.77877919166242687809943407586, 3.07280062367362272739357168848, 3.63834738003612573374401987901, 4.13392715776059607618726701998, 4.56164670331893563799858427284, 5.11112619081516648838162756916, 5.51398481501151096020107827863, 5.84411902691293804897692935056, 6.66366405066513570604633639399, 6.74483944259291106520329136216, 7.08543005360257846533748695637, 7.67967462989094291475036555323, 7.922361300288781335385497222834, 8.405355951009289857382128446472, 8.894118974858806846509640377690, 9.188121844361043554238643821877, 9.658046576257505396066782608257

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