Properties

Degree $4$
Conductor $1016064$
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  + 6·5-s − 7-s − 6·11-s + 12·17-s + 7·19-s + 19·25-s + 5·31-s − 6·35-s − 37-s + 6·47-s − 6·49-s − 36·55-s − 3·67-s + 15·73-s + 6·77-s + 27·79-s − 12·83-s + 72·85-s − 12·89-s + 42·95-s + 6·101-s + 5·103-s − 12·107-s − 5·109-s − 12·113-s − 12·119-s + 13·121-s + ⋯
L(s)  = 1  + 2.68·5-s − 0.377·7-s − 1.80·11-s + 2.91·17-s + 1.60·19-s + 19/5·25-s + 0.898·31-s − 1.01·35-s − 0.164·37-s + 0.875·47-s − 6/7·49-s − 4.85·55-s − 0.366·67-s + 1.75·73-s + 0.683·77-s + 3.03·79-s − 1.31·83-s + 7.80·85-s − 1.27·89-s + 4.30·95-s + 0.597·101-s + 0.492·103-s − 1.16·107-s − 0.478·109-s − 1.12·113-s − 1.10·119-s + 1.18·121-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: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1008} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1016064,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.867709390\)
\(L(\frac12)\) \(\approx\) \(3.867709390\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 27 T + 322 T^{2} - 27 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−10.00618018602297004553007880282, −9.837652035559882592675033631905, −9.484821166444071539078997288989, −9.299750802981559027408933590020, −8.513528867911589940564655701781, −8.013045436342747691905712330580, −7.69510935076730065069043461413, −7.33547611781743160015597808384, −6.60243585751128372839360272086, −6.24230118974771331470176126924, −5.71716745923910514169809733608, −5.48892943691590873712993826794, −5.14740552790729556275239509453, −4.95335560551707424016632813275, −3.71641947114840228992975792657, −3.19548452408356376078755964195, −2.69059908666722207005943690410, −2.38012340947232393790553933578, −1.46824724639271500189539877841, −1.00678612609309744881587395639, 1.00678612609309744881587395639, 1.46824724639271500189539877841, 2.38012340947232393790553933578, 2.69059908666722207005943690410, 3.19548452408356376078755964195, 3.71641947114840228992975792657, 4.95335560551707424016632813275, 5.14740552790729556275239509453, 5.48892943691590873712993826794, 5.71716745923910514169809733608, 6.24230118974771331470176126924, 6.60243585751128372839360272086, 7.33547611781743160015597808384, 7.69510935076730065069043461413, 8.013045436342747691905712330580, 8.513528867911589940564655701781, 9.299750802981559027408933590020, 9.484821166444071539078997288989, 9.837652035559882592675033631905, 10.00618018602297004553007880282

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