Properties

Label 4-39e4-1.1-c3e2-0-2
Degree $4$
Conductor $2313441$
Sign $1$
Analytic cond. $8053.60$
Root an. cond. $9.47322$
Motivic weight $3$
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  − 13·4-s + 105·16-s + 234·17-s − 156·23-s − 247·25-s + 282·29-s − 208·43-s − 494·49-s − 186·53-s + 290·61-s − 533·64-s − 3.04e3·68-s + 2.55e3·79-s + 2.02e3·92-s + 3.21e3·100-s + 858·101-s − 364·103-s + 3.01e3·107-s + 1.37e3·113-s − 3.66e3·116-s − 2.47e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.62·4-s + 1.64·16-s + 3.33·17-s − 1.41·23-s − 1.97·25-s + 1.80·29-s − 0.737·43-s − 1.44·49-s − 0.482·53-s + 0.608·61-s − 1.04·64-s − 5.42·68-s + 3.63·79-s + 2.29·92-s + 3.21·100-s + 0.845·101-s − 0.348·103-s + 2.72·107-s + 1.14·113-s − 2.93·116-s − 1.85·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2313441\)    =    \(3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(8053.60\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1521} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2313441,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.365073391\)
\(L(\frac12)\) \(\approx\) \(2.365073391\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
13 \( 1 \)
good2$C_2^2$ \( 1 + 13 T^{2} + p^{6} T^{4} \)
5$C_2^2$ \( 1 + 247 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 + 494 T^{2} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 2470 T^{2} + p^{6} T^{4} \)
17$C_2$ \( ( 1 - 117 T + p^{3} T^{2} )^{2} \)
19$C_2^2$ \( 1 + 650 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 + 78 T + p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 - 141 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 35282 T^{2} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 80639 T^{2} + p^{6} T^{4} \)
41$C_2^2$ \( 1 + 63895 T^{2} + p^{6} T^{4} \)
43$C_2$ \( ( 1 + 104 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 116818 T^{2} + p^{6} T^{4} \)
53$C_2$ \( ( 1 + 93 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 330070 T^{2} + p^{6} T^{4} \)
61$C_2$ \( ( 1 - 145 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 16822 T^{2} + p^{6} T^{4} \)
71$C_2^2$ \( 1 - 400478 T^{2} + p^{6} T^{4} \)
73$C_2^2$ \( 1 + 567359 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 1276 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 519766 T^{2} + p^{6} T^{4} \)
89$C_2^2$ \( 1 + 455650 T^{2} + p^{6} T^{4} \)
97$C_2^2$ \( 1 + 1784978 T^{2} + p^{6} T^{4} \)
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

−9.327486932291249285724095862797, −9.040852406691120715305276075180, −8.206063465790956996325082526317, −8.189244979245667943466975715228, −7.77423134477657699827200500138, −7.75316063070374836273976246608, −6.83742857468351378204704595839, −6.35093014288600729141919116372, −5.77759699241434142784358207487, −5.70450125610762439733533555459, −5.02172039165984458568670852084, −4.87402652743255754463629027390, −4.16889766828211972020294764055, −3.82991968795437775432555786804, −3.22096572160771695183392359924, −3.21004743394228165398118068961, −2.03790170726959255520540796148, −1.57727971356795834644781448684, −0.68555257622848918656260438554, −0.57034227746350611523180720085, 0.57034227746350611523180720085, 0.68555257622848918656260438554, 1.57727971356795834644781448684, 2.03790170726959255520540796148, 3.21004743394228165398118068961, 3.22096572160771695183392359924, 3.82991968795437775432555786804, 4.16889766828211972020294764055, 4.87402652743255754463629027390, 5.02172039165984458568670852084, 5.70450125610762439733533555459, 5.77759699241434142784358207487, 6.35093014288600729141919116372, 6.83742857468351378204704595839, 7.75316063070374836273976246608, 7.77423134477657699827200500138, 8.189244979245667943466975715228, 8.206063465790956996325082526317, 9.040852406691120715305276075180, 9.327486932291249285724095862797

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