Properties

Label 4-1840e2-1.1-c1e2-0-4
Degree $4$
Conductor $3385600$
Sign $1$
Analytic cond. $215.868$
Root an. cond. $3.83307$
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  + 4·5-s + 2·9-s − 8·19-s + 11·25-s + 18·29-s + 6·31-s + 18·41-s + 8·45-s + 5·49-s + 18·59-s − 4·61-s + 26·71-s − 4·79-s − 5·81-s + 20·89-s − 32·95-s − 26·101-s + 4·109-s − 22·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 72·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1.78·5-s + 2/3·9-s − 1.83·19-s + 11/5·25-s + 3.34·29-s + 1.07·31-s + 2.81·41-s + 1.19·45-s + 5/7·49-s + 2.34·59-s − 0.512·61-s + 3.08·71-s − 0.450·79-s − 5/9·81-s + 2.11·89-s − 3.28·95-s − 2.58·101-s + 0.383·109-s − 2·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.97·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3385600\)    =    \(2^{8} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(215.868\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3385600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.551777782\)
\(L(\frac12)\) \(\approx\) \(4.551777782\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 4 T + p T^{2} \)
23$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 57 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−9.891853816327565069451934964429, −9.106880328259528099367033204138, −8.583204717326241163457665832685, −8.454109571540342954702242546988, −8.014136237444527536420658788535, −7.41921580403427661255451879341, −6.86338147016201561933453114650, −6.55897312714781837698762306884, −6.20508359996827198648627932013, −6.14661255766429408260920904217, −5.27769580530086317560682466396, −5.13324573187851431949735853493, −4.36357284262363655520652507468, −4.36214722805439604445009313855, −3.63720928975287075157645829245, −2.69073401660752880830005209203, −2.44472970356902299281897618415, −2.24498697136300666590928618800, −1.16587171903868198005473538032, −0.947255169095409542204540926973, 0.947255169095409542204540926973, 1.16587171903868198005473538032, 2.24498697136300666590928618800, 2.44472970356902299281897618415, 2.69073401660752880830005209203, 3.63720928975287075157645829245, 4.36214722805439604445009313855, 4.36357284262363655520652507468, 5.13324573187851431949735853493, 5.27769580530086317560682466396, 6.14661255766429408260920904217, 6.20508359996827198648627932013, 6.55897312714781837698762306884, 6.86338147016201561933453114650, 7.41921580403427661255451879341, 8.014136237444527536420658788535, 8.454109571540342954702242546988, 8.583204717326241163457665832685, 9.106880328259528099367033204138, 9.891853816327565069451934964429

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