Properties

Label 4-20e4-1.1-c2e2-0-4
Degree $4$
Conductor $160000$
Sign $1$
Analytic cond. $118.792$
Root an. cond. $3.30139$
Motivic weight $2$
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  − 4·3-s + 4·7-s + 8·9-s + 16·11-s − 6·13-s − 14·17-s − 16·21-s − 4·23-s − 36·27-s − 104·31-s − 64·33-s + 6·37-s + 24·39-s − 16·41-s − 84·43-s − 36·47-s + 8·49-s + 56·51-s − 106·53-s − 96·61-s + 32·63-s + 124·67-s + 16·69-s + 56·71-s + 94·73-s + 64·77-s + 143·81-s + ⋯
L(s)  = 1  − 4/3·3-s + 4/7·7-s + 8/9·9-s + 1.45·11-s − 0.461·13-s − 0.823·17-s − 0.761·21-s − 0.173·23-s − 4/3·27-s − 3.35·31-s − 1.93·33-s + 6/37·37-s + 8/13·39-s − 0.390·41-s − 1.95·43-s − 0.765·47-s + 8/49·49-s + 1.09·51-s − 2·53-s − 1.57·61-s + 0.507·63-s + 1.85·67-s + 0.231·69-s + 0.788·71-s + 1.28·73-s + 0.831·77-s + 1.76·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(160000\)    =    \(2^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(118.792\)
Root analytic conductor: \(3.30139\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 160000,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7809207978\)
\(L(\frac12)\) \(\approx\) \(0.7809207978\)
\(L(2)\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \)
7$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \)
17$C_2$ \( ( 1 - 16 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} ) \)
19$C_2^2$ \( 1 - 322 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2$ \( ( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} ) \)
31$C_2$ \( ( 1 + 52 T + p^{2} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \)
41$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 84 T + 3528 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \)
47$C_2^2$ \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \)
59$C_2^2$ \( 1 - 6562 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 48 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 124 T + 7688 T^{2} - 124 p^{2} T^{3} + p^{4} T^{4} \)
71$C_2$ \( ( 1 - 28 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 94 T + 4418 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
83$C_2^2$ \( 1 - 36 T + 648 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 9442 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 126 T + 7938 T^{2} - 126 p^{2} T^{3} + p^{4} T^{4} \)
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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

−11.36219301202110009582046686494, −11.02925757659418387936956820616, −10.62568598103755953516354118666, −9.762342916850009284525604533544, −9.694841157114394839294440425383, −8.874035414996817492955701019736, −8.858393717482030118473741855376, −7.72371721650496784625870505845, −7.69332304342386810225971950094, −6.81494813700288390885837981447, −6.58802317176632074032570216965, −6.06944491546468071274347614546, −5.50709431715243810081971644691, −4.88041120016805143186330477747, −4.71301936461890810315053742308, −3.63743919069598103873364085530, −3.56679001993819847741093433042, −1.86793173718946404323648141080, −1.79004319827305602227911783764, −0.41548441986364967330235315004, 0.41548441986364967330235315004, 1.79004319827305602227911783764, 1.86793173718946404323648141080, 3.56679001993819847741093433042, 3.63743919069598103873364085530, 4.71301936461890810315053742308, 4.88041120016805143186330477747, 5.50709431715243810081971644691, 6.06944491546468071274347614546, 6.58802317176632074032570216965, 6.81494813700288390885837981447, 7.69332304342386810225971950094, 7.72371721650496784625870505845, 8.858393717482030118473741855376, 8.874035414996817492955701019736, 9.694841157114394839294440425383, 9.762342916850009284525604533544, 10.62568598103755953516354118666, 11.02925757659418387936956820616, 11.36219301202110009582046686494

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