Properties

Label 4-1280e2-1.1-c1e2-0-59
Degree $4$
Conductor $1638400$
Sign $1$
Analytic cond. $104.465$
Root an. cond. $3.19700$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·5-s + 2·7-s + 2·9-s + 6·13-s + 8·15-s − 6·17-s − 12·19-s − 4·21-s + 6·23-s + 11·25-s − 6·27-s − 8·35-s + 6·37-s − 12·39-s − 12·41-s − 6·43-s − 8·45-s − 18·47-s + 2·49-s + 12·51-s + 10·53-s + 24·57-s − 20·59-s − 24·61-s + 4·63-s − 24·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s + 0.755·7-s + 2/3·9-s + 1.66·13-s + 2.06·15-s − 1.45·17-s − 2.75·19-s − 0.872·21-s + 1.25·23-s + 11/5·25-s − 1.15·27-s − 1.35·35-s + 0.986·37-s − 1.92·39-s − 1.87·41-s − 0.914·43-s − 1.19·45-s − 2.62·47-s + 2/7·49-s + 1.68·51-s + 1.37·53-s + 3.17·57-s − 2.60·59-s − 3.07·61-s + 0.503·63-s − 2.97·65-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(1638400\)    =    \(2^{16} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(104.465\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1280} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1638400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
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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

−9.136686175933935856831892765616, −9.021090783569475777108770615283, −8.598880242754077087487523627891, −8.175757954531525181568053823565, −8.008204001041024588045940602359, −7.41941177266018380740119629381, −6.81277547351238680714028306395, −6.65479844038999291319325866156, −6.07687484276582009373022880009, −5.97867336419372726990818446053, −4.87242873124890504106171349912, −4.81005461184919266983973023639, −4.34109812363942251518470323405, −4.06748007652875230407350355369, −3.36372373265683444113296474755, −2.92377876738897558956302032812, −1.76630655363286569743250153135, −1.44174896761086195869964540132, 0, 0, 1.44174896761086195869964540132, 1.76630655363286569743250153135, 2.92377876738897558956302032812, 3.36372373265683444113296474755, 4.06748007652875230407350355369, 4.34109812363942251518470323405, 4.81005461184919266983973023639, 4.87242873124890504106171349912, 5.97867336419372726990818446053, 6.07687484276582009373022880009, 6.65479844038999291319325866156, 6.81277547351238680714028306395, 7.41941177266018380740119629381, 8.008204001041024588045940602359, 8.175757954531525181568053823565, 8.598880242754077087487523627891, 9.021090783569475777108770615283, 9.136686175933935856831892765616

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