Properties

Label 4-38e4-1.1-c0e2-0-0
Degree $4$
Conductor $2085136$
Sign $1$
Analytic cond. $0.519336$
Root an. cond. $0.848910$
Motivic weight $0$
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 − 9-s − 2·11-s + 17-s − 2·23-s + 25-s − 2·35-s + 43-s − 45-s + 47-s + 49-s − 2·55-s + 61-s + 2·63-s + 73-s + 4·77-s + 4·83-s + 85-s + 2·99-s − 2·101-s − 2·115-s − 2·119-s + 121-s + 2·125-s + 127-s + 131-s + ⋯
L(s)  = 1  + 5-s − 2·7-s − 9-s − 2·11-s + 17-s − 2·23-s + 25-s − 2·35-s + 43-s − 45-s + 47-s + 49-s − 2·55-s + 61-s + 2·63-s + 73-s + 4·77-s + 4·83-s + 85-s + 2·99-s − 2·101-s − 2·115-s − 2·119-s + 121-s + 2·125-s + 127-s + 131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2085136\)    =    \(2^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(0.519336\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1444} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2085136,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6377214842\)
\(L(\frac12)\) \(\approx\) \(0.6377214842\)
\(L(1)\) 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 \( 1 \)
19 \( 1 \)
good3$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
7$C_2$ \( ( 1 + T + T^{2} )^{2} \)
11$C_2$ \( ( 1 + T + T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
23$C_2$ \( ( 1 + T + T^{2} )^{2} \)
29$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
47$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_1$ \( ( 1 - T )^{4} \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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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

−10.07357511585902444549712042577, −9.450894102198111153581618043208, −9.378425387187765761140308637221, −8.829971732467341798668091817329, −8.159199594605645473307960865990, −8.013120238370716009915483029140, −7.64468821262505167670420497366, −6.94543881930385196867850818075, −6.61236037191190504224677213673, −6.08622087369974846490588521189, −5.86864876691273257795639999140, −5.45530975981446613030679895909, −5.23553128078275267156677108279, −4.47181732868480573484273247992, −3.76164804416741179238826528102, −3.28912761803733197402161722223, −2.91333630617745562481150830755, −2.40290751585530755382738396254, −2.04427352570763004950562669902, −0.61502814383840666277453031149, 0.61502814383840666277453031149, 2.04427352570763004950562669902, 2.40290751585530755382738396254, 2.91333630617745562481150830755, 3.28912761803733197402161722223, 3.76164804416741179238826528102, 4.47181732868480573484273247992, 5.23553128078275267156677108279, 5.45530975981446613030679895909, 5.86864876691273257795639999140, 6.08622087369974846490588521189, 6.61236037191190504224677213673, 6.94543881930385196867850818075, 7.64468821262505167670420497366, 8.013120238370716009915483029140, 8.159199594605645473307960865990, 8.829971732467341798668091817329, 9.378425387187765761140308637221, 9.450894102198111153581618043208, 10.07357511585902444549712042577

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