Properties

Label 4-3920e2-1.1-c0e2-0-4
Degree $4$
Conductor $15366400$
Sign $1$
Analytic cond. $3.82724$
Root an. cond. $1.39869$
Motivic weight $0$
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  + 2·5-s + 9-s + 3·25-s − 2·29-s − 2·41-s + 2·45-s − 2·61-s + 2·89-s + 2·101-s + 2·109-s + 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 2·5-s + 9-s + 3·25-s − 2·29-s − 2·41-s + 2·45-s − 2·61-s + 2·89-s + 2·101-s + 2·109-s + 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15366400\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(3.82724\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3920} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15366400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.483664859\)
\(L(\frac12)\) \(\approx\) \(2.483664859\)
\(L(1)\) 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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_2$ \( ( 1 + T + T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_2$ \( ( 1 + T + T^{2} )^{2} \)
43$C_2^2$ \( 1 - T^{2} + T^{4} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T + T^{2} )^{2} \)
67$C_2^2$ \( 1 - T^{2} + T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 - T^{2} + T^{4} \)
89$C_2$ \( ( 1 - T + T^{2} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−8.956108210562206209990784044381, −8.601377427770113582300161973016, −8.128725964837760640838111438724, −7.59514839199470243006805898537, −7.33118290207869700134588603143, −6.96165348348636394809829758425, −6.57040191620155999160882826952, −6.26634060818155616380010169127, −5.72001267061123211093105438507, −5.70779353000963306177534561303, −5.07946398376705954394716485155, −4.75322985178008914011936985210, −4.46792955475968031660661767907, −3.79898656175543985972988257280, −3.18649694421326464983934446220, −3.12591970725538270566900339192, −2.14305379732536993395158047610, −1.91424914110462113324793165761, −1.68155899929199723058289274410, −0.915010044279353065802582012378, 0.915010044279353065802582012378, 1.68155899929199723058289274410, 1.91424914110462113324793165761, 2.14305379732536993395158047610, 3.12591970725538270566900339192, 3.18649694421326464983934446220, 3.79898656175543985972988257280, 4.46792955475968031660661767907, 4.75322985178008914011936985210, 5.07946398376705954394716485155, 5.70779353000963306177534561303, 5.72001267061123211093105438507, 6.26634060818155616380010169127, 6.57040191620155999160882826952, 6.96165348348636394809829758425, 7.33118290207869700134588603143, 7.59514839199470243006805898537, 8.128725964837760640838111438724, 8.601377427770113582300161973016, 8.956108210562206209990784044381

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