Properties

Label 4-1008e2-1.1-c1e2-0-6
Degree $4$
Conductor $1016064$
Sign $1$
Analytic cond. $64.7851$
Root an. cond. $2.83706$
Motivic weight $1$
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 − 7-s − 2·11-s − 6·13-s + 8·17-s − 19-s − 8·23-s + 5·25-s − 8·29-s + 3·31-s + 2·35-s + 37-s − 12·41-s − 22·43-s − 6·47-s − 6·49-s − 12·53-s + 4·55-s − 4·59-s + 6·61-s + 12·65-s + 13·67-s − 20·71-s + 11·73-s + 2·77-s − 3·79-s + 4·83-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s − 0.603·11-s − 1.66·13-s + 1.94·17-s − 0.229·19-s − 1.66·23-s + 25-s − 1.48·29-s + 0.538·31-s + 0.338·35-s + 0.164·37-s − 1.87·41-s − 3.35·43-s − 0.875·47-s − 6/7·49-s − 1.64·53-s + 0.539·55-s − 0.520·59-s + 0.768·61-s + 1.48·65-s + 1.58·67-s − 2.37·71-s + 1.28·73-s + 0.227·77-s − 0.337·79-s + 0.439·83-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(64.7851\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1016064,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3765024446\)
\(L(\frac12)\) \(\approx\) \(0.3765024446\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−10.12439743185892429556539573398, −9.860708935230116796945196939023, −9.637483483162028476701941310990, −8.826926585786311223434476879180, −8.339207801866229059117350340323, −8.052555698255183327580179867984, −7.75588840069431215917490103227, −7.25708377973657127048374765194, −7.01396122239580486441216968313, −6.18445510341878905351390689953, −6.11669449839871372112995610996, −5.08873450902929005924555282010, −5.04526338719711150937495835419, −4.67771826047983588679405632832, −3.70422100998641008566470191326, −3.35769667524568380982176608366, −3.14459992634473508449514154849, −2.14243528799439217826877084976, −1.65325479673313526745482323510, −0.27063631687424285128476980581, 0.27063631687424285128476980581, 1.65325479673313526745482323510, 2.14243528799439217826877084976, 3.14459992634473508449514154849, 3.35769667524568380982176608366, 3.70422100998641008566470191326, 4.67771826047983588679405632832, 5.04526338719711150937495835419, 5.08873450902929005924555282010, 6.11669449839871372112995610996, 6.18445510341878905351390689953, 7.01396122239580486441216968313, 7.25708377973657127048374765194, 7.75588840069431215917490103227, 8.052555698255183327580179867984, 8.339207801866229059117350340323, 8.826926585786311223434476879180, 9.637483483162028476701941310990, 9.860708935230116796945196939023, 10.12439743185892429556539573398

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