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Properties

Label 4-1008e2-1.1-c0e2-0-1
Degree $4$
Conductor $1016064$
Sign $1$
Analytic cond. $0.253066$
Root an. cond. $0.709265$
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  − 4-s + 2·11-s + 16-s + 2·29-s − 2·37-s + 2·43-s − 2·44-s − 49-s − 2·53-s − 64-s + 2·67-s − 2·107-s + 2·109-s − 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·172-s + 173-s + ⋯
L(s)  = 1  − 4-s + 2·11-s + 16-s + 2·29-s − 2·37-s + 2·43-s − 2·44-s − 49-s − 2·53-s − 64-s + 2·67-s − 2·107-s + 2·109-s − 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·172-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.253066\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1016064,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8974010899\)
\(L(\frac12)\) \(\approx\) \(0.8974010899\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good5$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
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

−10.16528615582161488930150417818, −9.967813681169342208175029349838, −9.437161047326423074733404257214, −9.127131572802656803559456188956, −8.771656860453085751454923820629, −8.506134317780769892805558816464, −7.85035667573313045575570778239, −7.65611866244347104797648494049, −6.72230448940592274129180487575, −6.66834373646425384122919679296, −6.24167129196224017674919038079, −5.61201941496033626014267506117, −5.05738762063776045952704665487, −4.70368051167754649848998230011, −4.14426234420912151412183513908, −3.76672239131288494893803170190, −3.30585615892353791783724864143, −2.59611604592398391046324998051, −1.60613489739121172351148783062, −1.05803658760874879022386712685, 1.05803658760874879022386712685, 1.60613489739121172351148783062, 2.59611604592398391046324998051, 3.30585615892353791783724864143, 3.76672239131288494893803170190, 4.14426234420912151412183513908, 4.70368051167754649848998230011, 5.05738762063776045952704665487, 5.61201941496033626014267506117, 6.24167129196224017674919038079, 6.66834373646425384122919679296, 6.72230448940592274129180487575, 7.65611866244347104797648494049, 7.85035667573313045575570778239, 8.506134317780769892805558816464, 8.771656860453085751454923820629, 9.127131572802656803559456188956, 9.437161047326423074733404257214, 9.967813681169342208175029349838, 10.16528615582161488930150417818

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