Properties

Label 4-1260e2-1.1-c0e2-0-2
Degree $4$
Conductor $1587600$
Sign $1$
Analytic cond. $0.395417$
Root an. cond. $0.792982$
Motivic weight $0$
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  − 3-s − 5-s − 7-s + 11-s − 2·13-s + 15-s + 4·17-s + 21-s + 27-s + 29-s − 33-s + 35-s + 2·39-s + 47-s − 4·51-s − 55-s + 2·65-s − 2·71-s − 2·73-s − 77-s + 79-s − 81-s + 83-s − 4·85-s − 87-s + 2·91-s + 97-s + ⋯
L(s)  = 1  − 3-s − 5-s − 7-s + 11-s − 2·13-s + 15-s + 4·17-s + 21-s + 27-s + 29-s − 33-s + 35-s + 2·39-s + 47-s − 4·51-s − 55-s + 2·65-s − 2·71-s − 2·73-s − 77-s + 79-s − 81-s + 83-s − 4·85-s − 87-s + 2·91-s + 97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1587600\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.395417\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1587600,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4967989111\)
\(L(\frac12)\) \(\approx\) \(0.4967989111\)
\(L(1)\) 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$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + T + T^{2} \)
good11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 + T + T^{2} )^{2} \)
17$C_1$ \( ( 1 - T )^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
43$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
47$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 + T + T^{2} )^{2} \)
73$C_2$ \( ( 1 + T + T^{2} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
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

−10.06730644495767293612875970793, −9.893887134316449520003900315330, −9.348551761298826515776756762890, −9.041205284071703107447268688878, −8.315673376621380181281186083999, −7.959218325176305146960312070330, −7.54736716130597860415563334494, −7.20111279648771999587577018523, −6.98078418398655709014225964889, −6.12070785493283817583891144665, −6.00862089402620994796524798995, −5.57621709765312417170182597516, −4.86853994502728291245196181917, −4.83677616947501216250468290792, −3.91042395338467591995970446413, −3.60742640802207272197799622227, −2.97895615242959328264702728955, −2.75524952987346086353900111449, −1.45142171725875049176399491500, −0.70555636344158107069093094353, 0.70555636344158107069093094353, 1.45142171725875049176399491500, 2.75524952987346086353900111449, 2.97895615242959328264702728955, 3.60742640802207272197799622227, 3.91042395338467591995970446413, 4.83677616947501216250468290792, 4.86853994502728291245196181917, 5.57621709765312417170182597516, 6.00862089402620994796524798995, 6.12070785493283817583891144665, 6.98078418398655709014225964889, 7.20111279648771999587577018523, 7.54736716130597860415563334494, 7.959218325176305146960312070330, 8.315673376621380181281186083999, 9.041205284071703107447268688878, 9.348551761298826515776756762890, 9.893887134316449520003900315330, 10.06730644495767293612875970793

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