Properties

Label 4-20e4-1.1-c5e2-0-8
Degree $4$
Conductor $160000$
Sign $1$
Analytic cond. $4115.67$
Root an. cond. $8.00958$
Motivic weight $5$
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  + 470·9-s + 296·11-s + 2.12e3·19-s + 6.82e3·29-s + 4.89e3·31-s − 1.87e4·41-s − 3.25e3·49-s − 4.00e4·59-s + 6.46e4·61-s + 6.52e4·71-s − 6.67e4·79-s + 1.61e5·81-s − 2.02e5·89-s + 1.39e5·99-s − 1.79e5·101-s − 7.36e4·109-s − 2.56e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6.60e5·169-s + ⋯
L(s)  = 1  + 1.93·9-s + 0.737·11-s + 1.34·19-s + 1.50·29-s + 0.915·31-s − 1.74·41-s − 0.193·49-s − 1.49·59-s + 2.22·61-s + 1.53·71-s − 1.20·79-s + 2.74·81-s − 2.71·89-s + 1.42·99-s − 1.75·101-s − 0.593·109-s − 1.59·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.77·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(160000\)    =    \(2^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(4115.67\)
Root analytic conductor: \(8.00958\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 160000,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(4.714841461\)
\(L(\frac12)\) \(\approx\) \(4.714841461\)
\(L(\frac{7}{2})\) 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$C_2^2$ \( 1 - 470 T^{2} + p^{10} T^{4} \)
7$C_2^2$ \( 1 + 3250 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 - 148 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 3910 p^{2} T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 - 24030 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 - 1060 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 4016110 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 - 3410 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2448 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 138654790 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 + 9398 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 292469350 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 312570270 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 267759270 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 + 20020 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 32302 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 1017334570 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 32648 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 2642720110 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 + 33360 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 7598656630 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 + 101370 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 3004635070 T^{2} + p^{10} T^{4} \)
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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

−10.63711314705327424189992972677, −10.02358959797624898560643994627, −9.826582448689659733834801006543, −9.564058936778676808309358684555, −8.904592511014022302758551925819, −8.229087475235888577866423995318, −8.056006419736228498335078704124, −7.26370010723787706581985549315, −6.85172931810763192178561952344, −6.71575578682867340735517279626, −6.02540750850544426033913283284, −5.18436907243529206725369071465, −4.91703326635284201664069702733, −4.20066058815643123634873043310, −3.88356738675044604493213960324, −3.15400725806146634871805935912, −2.51317803496221983519859931273, −1.42773789144470423649753859643, −1.37770476687946560940805465700, −0.58035348156098161694013137955, 0.58035348156098161694013137955, 1.37770476687946560940805465700, 1.42773789144470423649753859643, 2.51317803496221983519859931273, 3.15400725806146634871805935912, 3.88356738675044604493213960324, 4.20066058815643123634873043310, 4.91703326635284201664069702733, 5.18436907243529206725369071465, 6.02540750850544426033913283284, 6.71575578682867340735517279626, 6.85172931810763192178561952344, 7.26370010723787706581985549315, 8.056006419736228498335078704124, 8.229087475235888577866423995318, 8.904592511014022302758551925819, 9.564058936778676808309358684555, 9.826582448689659733834801006543, 10.02358959797624898560643994627, 10.63711314705327424189992972677

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