Properties

Label 4-20e4-1.1-c5e2-0-4
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

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 90·9-s − 264·11-s + 1.00e3·19-s − 4.38e3·29-s − 4.62e3·31-s + 2.48e3·41-s + 4.03e3·49-s + 1.59e4·59-s + 3.32e4·61-s + 4.90e4·71-s − 9.24e4·79-s − 5.09e4·81-s + 2.20e5·89-s + 2.37e4·99-s + 2.83e5·101-s + 3.49e5·109-s − 2.69e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.52e5·169-s + ⋯
L(s)  = 1  − 0.370·9-s − 0.657·11-s + 0.635·19-s − 0.967·29-s − 0.864·31-s + 0.230·41-s + 0.239·49-s + 0.596·59-s + 1.14·61-s + 1.15·71-s − 1.66·79-s − 0.862·81-s + 2.95·89-s + 0.243·99-s + 2.76·101-s + 2.81·109-s − 1.67·121-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 − 0.410·169-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 160000,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.769960301\)
\(L(\frac12)\) \(\approx\) \(1.769960301\)
\(L(\frac{7}{2})\) 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$C_2^2$ \( 1 + 10 p^{2} T^{2} + p^{10} T^{4} \)
7$C_2^2$ \( 1 - 4030 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 + 12 p T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 152330 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 - 2790430 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 - 500 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 170590 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 + 2190 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2312 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 12305350 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 - 1242 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 131332490 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 415288270 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 392614630 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 - 7980 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 16622 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 2696981350 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 24528 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3726958510 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 + 46240 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 5217997510 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 - 110310 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 11030942590 T^{2} + p^{10} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.20126901418439867195428815207, −10.11774491610350593784733003098, −9.935665805690466314738188326736, −9.260495622751062300999562447070, −8.838173691090341364202998843199, −8.504310067211096401092354883041, −7.79526988517850202405748539369, −7.43964408757535550862562883538, −7.14264599034062332950794760910, −6.28528227361147548101432668806, −5.97868768977696243609796420838, −5.22499422355608568090279560161, −5.12393732157512081686439900406, −4.27308816015194153223599460256, −3.61203156203784594417808070061, −3.22064386513993205444521383317, −2.38211939212763706650181030381, −1.97227214758327997329873411582, −1.03103811712080657343912961905, −0.36712056391158025581312131368, 0.36712056391158025581312131368, 1.03103811712080657343912961905, 1.97227214758327997329873411582, 2.38211939212763706650181030381, 3.22064386513993205444521383317, 3.61203156203784594417808070061, 4.27308816015194153223599460256, 5.12393732157512081686439900406, 5.22499422355608568090279560161, 5.97868768977696243609796420838, 6.28528227361147548101432668806, 7.14264599034062332950794760910, 7.43964408757535550862562883538, 7.79526988517850202405748539369, 8.504310067211096401092354883041, 8.838173691090341364202998843199, 9.260495622751062300999562447070, 9.935665805690466314738188326736, 10.11774491610350593784733003098, 11.20126901418439867195428815207

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