Properties

Label 4-1280e2-1.1-c0e2-0-1
Degree $4$
Conductor $1638400$
Sign $1$
Analytic cond. $0.408069$
Root an. cond. $0.799251$
Motivic weight $0$
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  − 2·5-s + 3·25-s + 4·61-s − 81-s − 4·89-s − 4·109-s + 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 2·5-s + 3·25-s + 4·61-s − 81-s − 4·89-s − 4·109-s + 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1638400\)    =    \(2^{16} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.408069\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1638400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6325797592\)
\(L(\frac12)\) \(\approx\) \(0.6325797592\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$ \( ( 1 - T )^{4} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$ \( ( 1 + T )^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−9.989399383115153507899038229426, −9.726596552189099725140848056567, −9.202713195724998338614066340546, −8.664963403798591759347843478425, −8.295998040127648597193030973554, −8.184355127075122758496533876636, −7.73697928437011101205136340001, −6.99704207618037254120184822889, −6.99155893631541424883519734617, −6.70421426577398929925581216387, −5.71458161335562822697603802377, −5.51635208355382842977662266503, −4.88521014165427488349416963154, −4.40751705592858610362505647469, −3.86772964661671157802939162997, −3.82926753282081635255589313073, −2.92100914359994698244124040325, −2.71468546190314114139779289000, −1.65679114841356520587875148021, −0.71517719326576147399506352774, 0.71517719326576147399506352774, 1.65679114841356520587875148021, 2.71468546190314114139779289000, 2.92100914359994698244124040325, 3.82926753282081635255589313073, 3.86772964661671157802939162997, 4.40751705592858610362505647469, 4.88521014165427488349416963154, 5.51635208355382842977662266503, 5.71458161335562822697603802377, 6.70421426577398929925581216387, 6.99155893631541424883519734617, 6.99704207618037254120184822889, 7.73697928437011101205136340001, 8.184355127075122758496533876636, 8.295998040127648597193030973554, 8.664963403798591759347843478425, 9.202713195724998338614066340546, 9.726596552189099725140848056567, 9.989399383115153507899038229426

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