Properties

Label 4-1620e2-1.1-c3e2-0-8
Degree $4$
Conductor $2624400$
Sign $1$
Analytic cond. $9136.12$
Root an. cond. $9.77666$
Motivic weight $3$
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  + 5·5-s − 2·7-s + 30·11-s + 4·13-s − 180·17-s − 56·19-s + 120·23-s + 210·29-s + 4·31-s − 10·35-s + 400·37-s + 240·41-s + 136·43-s − 120·47-s + 343·49-s + 60·53-s + 150·55-s − 450·59-s + 166·61-s + 20·65-s − 908·67-s + 2.04e3·71-s − 500·73-s − 60·77-s + 916·79-s − 1.14e3·83-s − 900·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.107·7-s + 0.822·11-s + 0.0853·13-s − 2.56·17-s − 0.676·19-s + 1.08·23-s + 1.34·29-s + 0.0231·31-s − 0.0482·35-s + 1.77·37-s + 0.914·41-s + 0.482·43-s − 0.372·47-s + 49-s + 0.155·53-s + 0.367·55-s − 0.992·59-s + 0.348·61-s + 0.0381·65-s − 1.65·67-s + 3.40·71-s − 0.801·73-s − 0.0888·77-s + 1.30·79-s − 1.50·83-s − 1.14·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2624400\)    =    \(2^{4} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(9136.12\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2624400,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.400178093\)
\(L(\frac12)\) \(\approx\) \(4.400178093\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 - p T + p^{2} T^{2} \)
good7$C_2^2$ \( 1 + 2 T - 339 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 30 T - 431 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2^2$ \( 1 - 4 T - 2181 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2$ \( ( 1 + 90 T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 + 28 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 120 T + 2233 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 - 210 T + 19711 T^{2} - 210 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2^2$ \( 1 - 4 T - 29775 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2$ \( ( 1 - 200 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 240 T - 11321 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 - 136 T - 61011 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 + 120 T - 89423 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2$ \( ( 1 - 30 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 450 T - 2879 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 166 T - 199425 T^{2} - 166 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 908 T + 523701 T^{2} + 908 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 1020 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 250 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 916 T + 346017 T^{2} - 916 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2^2$ \( 1 + 1140 T + 727813 T^{2} + 1140 p^{3} T^{3} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 420 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 + 1538 T + 1452771 T^{2} + 1538 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.179481513238180583526154089039, −8.883759943364357859218304927373, −8.446319392553720728857708284538, −8.303412213753716073593860681159, −7.49537581278123381593460022759, −7.20050436109476826106362353364, −6.69634879381948113470005802642, −6.43978269491275225685642895930, −6.10246965044580990215061873354, −5.73808956093631785789058529017, −4.90673438356624743480239432816, −4.68188998467233301478501893869, −4.17237863640150590573124367733, −4.01123908595908201391263307822, −2.99694672027115655748303861153, −2.80725673286772482099290753093, −2.05587660198820689938867939007, −1.85016471102593187683118742496, −0.74324754200861791421041994190, −0.65143256734211370247081735996, 0.65143256734211370247081735996, 0.74324754200861791421041994190, 1.85016471102593187683118742496, 2.05587660198820689938867939007, 2.80725673286772482099290753093, 2.99694672027115655748303861153, 4.01123908595908201391263307822, 4.17237863640150590573124367733, 4.68188998467233301478501893869, 4.90673438356624743480239432816, 5.73808956093631785789058529017, 6.10246965044580990215061873354, 6.43978269491275225685642895930, 6.69634879381948113470005802642, 7.20050436109476826106362353364, 7.49537581278123381593460022759, 8.303412213753716073593860681159, 8.446319392553720728857708284538, 8.883759943364357859218304927373, 9.179481513238180583526154089039

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