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Properties

Label 4-740e2-1.1-c0e2-0-3
Degree $4$
Conductor $547600$
Sign $1$
Analytic cond. $0.136388$
Root an. cond. $0.607707$
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  − 2·3-s + 2·5-s + 9-s − 4·15-s + 3·25-s + 2·27-s + 2·31-s − 2·43-s + 2·45-s + 49-s − 2·61-s − 2·71-s + 2·73-s − 6·75-s + 2·79-s − 4·81-s − 2·89-s − 4·93-s + 2·97-s − 2·103-s − 2·109-s − 2·113-s + 121-s + 4·125-s + 127-s + 4·129-s + 131-s + ⋯
L(s)  = 1  − 2·3-s + 2·5-s + 9-s − 4·15-s + 3·25-s + 2·27-s + 2·31-s − 2·43-s + 2·45-s + 49-s − 2·61-s − 2·71-s + 2·73-s − 6·75-s + 2·79-s − 4·81-s − 2·89-s − 4·93-s + 2·97-s − 2·103-s − 2·109-s − 2·113-s + 121-s + 4·125-s + 127-s + 4·129-s + 131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(547600\)    =    \(2^{4} \cdot 5^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(0.136388\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 547600,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6089882636\)
\(L(\frac12)\) \(\approx\) \(0.6089882636\)
\(L(1)\) 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$C_1$ \( ( 1 - T )^{2} \)
37$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 + T + T^{2} )^{2} \)
7$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
41$C_2^2$ \( 1 - T^{2} + T^{4} \)
43$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
47$C_2^2$ \( 1 - T^{2} + T^{4} \)
53$C_2^2$ \( 1 - T^{2} + T^{4} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_2$ \( ( 1 + T + T^{2} )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
83$C_2^2$ \( 1 - T^{2} + T^{4} \)
89$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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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

−10.67516636882129806210701009476, −10.56862508602636943109469905733, −9.994769794085147461570745753155, −9.799286297463342101753300914211, −9.103515677188854980741298460547, −8.867974839052776531917143822463, −8.269091570514110769839769188153, −7.79575930962859574121482498576, −6.81138345199920516422076649275, −6.66478762581296740870360209884, −6.26784537709608327171777563953, −6.02213689077084962837281446081, −5.30374524985244088228701352809, −5.28798658406050219517083468675, −4.81288738167812688382663982630, −4.17011262634207891587118467124, −2.93856423426109703182556366949, −2.77139635544225333279026596199, −1.76449077401087519533179252045, −1.02748974401997445720603175672, 1.02748974401997445720603175672, 1.76449077401087519533179252045, 2.77139635544225333279026596199, 2.93856423426109703182556366949, 4.17011262634207891587118467124, 4.81288738167812688382663982630, 5.28798658406050219517083468675, 5.30374524985244088228701352809, 6.02213689077084962837281446081, 6.26784537709608327171777563953, 6.66478762581296740870360209884, 6.81138345199920516422076649275, 7.79575930962859574121482498576, 8.269091570514110769839769188153, 8.867974839052776531917143822463, 9.103515677188854980741298460547, 9.799286297463342101753300914211, 9.994769794085147461570745753155, 10.56862508602636943109469905733, 10.67516636882129806210701009476

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