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Properties

Label 4-2880e2-1.1-c0e2-0-4
Degree $4$
Conductor $8294400$
Sign $1$
Analytic cond. $2.06585$
Root an. cond. $1.19887$
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  + 2·13-s + 2·17-s − 25-s − 2·37-s + 2·53-s + 2·73-s + 2·97-s − 4·101-s + 2·113-s − 2·121-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 + ⋯
L(s)  = 1  + 2·13-s + 2·17-s − 25-s − 2·37-s + 2·53-s + 2·73-s + 2·97-s − 4·101-s + 2·113-s − 2·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8294400\)    =    \(2^{12} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2.06585\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8294400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.647709092\)
\(L(\frac12)\) \(\approx\) \(1.647709092\)
\(L(1)\) 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 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 + T^{4} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
17$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
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_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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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

−9.128242467671934271318311889394, −8.652418937786704961688267689441, −8.298600586249430583897257021157, −8.236205154807722927687270675691, −7.53301541283942546343126999309, −7.44195315300151544239062025821, −6.84099824503095548871437849223, −6.41902427963126725495844519235, −6.14375428429143603841538371904, −5.61163699486104695045117264255, −5.32354147009480676984849420385, −5.16568764777062778996810697081, −4.23837046942008213553214128633, −3.95216783203798512716692098249, −3.42529397274657834581252151177, −3.42414984652996659659461649182, −2.62481241911682814146097580694, −1.96688198360800939298263431269, −1.41376208015729608012874519350, −0.917948674155859498351291349617, 0.917948674155859498351291349617, 1.41376208015729608012874519350, 1.96688198360800939298263431269, 2.62481241911682814146097580694, 3.42414984652996659659461649182, 3.42529397274657834581252151177, 3.95216783203798512716692098249, 4.23837046942008213553214128633, 5.16568764777062778996810697081, 5.32354147009480676984849420385, 5.61163699486104695045117264255, 6.14375428429143603841538371904, 6.41902427963126725495844519235, 6.84099824503095548871437849223, 7.44195315300151544239062025821, 7.53301541283942546343126999309, 8.236205154807722927687270675691, 8.298600586249430583897257021157, 8.652418937786704961688267689441, 9.128242467671934271318311889394

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