Properties

Degree $4$
Conductor $64016001$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s + 2·7-s + 4·13-s + 6·17-s + 4·19-s − 12·23-s − 4·25-s + 4·28-s + 6·29-s + 16·31-s − 14·37-s − 6·41-s + 4·43-s + 24·47-s + 3·49-s + 8·52-s + 6·53-s + 4·61-s − 8·64-s + 4·67-s + 12·68-s − 12·71-s − 8·73-s + 8·76-s − 2·79-s − 12·83-s + 8·91-s + ⋯
L(s)  = 1  + 4-s + 0.755·7-s + 1.10·13-s + 1.45·17-s + 0.917·19-s − 2.50·23-s − 4/5·25-s + 0.755·28-s + 1.11·29-s + 2.87·31-s − 2.30·37-s − 0.937·41-s + 0.609·43-s + 3.50·47-s + 3/7·49-s + 1.10·52-s + 0.824·53-s + 0.512·61-s − 64-s + 0.488·67-s + 1.45·68-s − 1.42·71-s − 0.936·73-s + 0.917·76-s − 0.225·79-s − 1.31·83-s + 0.838·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64016001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64016001 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(64016001\)    =    \(3^{4} \cdot 7^{2} \cdot 127^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{8001} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 64016001,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.461122381\)
\(L(\frac12)\) \(\approx\) \(5.461122381\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
127$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 6 T + 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 6 T + 85 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 6 T + 91 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 112 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 120 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T - 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 12 T + 172 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 8 T + 108 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T + 135 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 10 T + 165 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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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

−8.011752777519427564079207414797, −7.59142499423268438437091792175, −7.43587425908044755501057476873, −6.93245116135414882076674201949, −6.73242015990030196977210017948, −6.03547288909840762799806781183, −5.99581673424060231592990485831, −5.83777342285728021511503054030, −5.13824239300944879145112804146, −5.08630175609699120559103937533, −4.21167849494724243594894881229, −4.16772098887322074841561776458, −3.78876775369904540773722427496, −3.24591463955147718577444817011, −2.80222178944816551373719985819, −2.46312038230056082614469831264, −2.03780219137573842424714449543, −1.36281015229573483665711493292, −1.28266928430766931766973504102, −0.55150585258363356747223626786, 0.55150585258363356747223626786, 1.28266928430766931766973504102, 1.36281015229573483665711493292, 2.03780219137573842424714449543, 2.46312038230056082614469831264, 2.80222178944816551373719985819, 3.24591463955147718577444817011, 3.78876775369904540773722427496, 4.16772098887322074841561776458, 4.21167849494724243594894881229, 5.08630175609699120559103937533, 5.13824239300944879145112804146, 5.83777342285728021511503054030, 5.99581673424060231592990485831, 6.03547288909840762799806781183, 6.73242015990030196977210017948, 6.93245116135414882076674201949, 7.43587425908044755501057476873, 7.59142499423268438437091792175, 8.011752777519427564079207414797

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