Properties

Label 4-60e4-1.1-c1e2-0-8
Degree $4$
Conductor $12960000$
Sign $1$
Analytic cond. $826.340$
Root an. cond. $5.36154$
Motivic weight $1$
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  − 12·11-s − 8·19-s + 12·29-s + 8·31-s + 10·49-s + 12·59-s + 4·61-s + 24·71-s − 8·79-s − 24·89-s − 12·101-s − 4·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 3.61·11-s − 1.83·19-s + 2.22·29-s + 1.43·31-s + 10/7·49-s + 1.56·59-s + 0.512·61-s + 2.84·71-s − 0.900·79-s − 2.54·89-s − 1.19·101-s − 0.383·109-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12960000\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(826.340\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.210256406\)
\(L(\frac12)\) \(\approx\) \(1.210256406\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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

−8.455247075133481006299115217412, −8.220724802447683652948001314610, −8.049278148922435234064451192165, −8.024423368007723938671909323732, −7.07565868062094417145466418044, −6.94788494901748043621824269016, −6.70572607022142048531350673876, −5.96426032676046222619033011623, −5.60122303682821828430940295754, −5.52522631346523749459457208012, −4.74090596514049221835074435982, −4.73529025010744677752477421678, −4.30164505802317307492007249569, −3.66836286189505578103644705043, −2.97296350566869200410881877135, −2.69294332589138691533521372601, −2.37794142916099814416965707949, −2.06962935183071438755445512505, −0.934917132768772682642972601934, −0.38450013899625558583774129159, 0.38450013899625558583774129159, 0.934917132768772682642972601934, 2.06962935183071438755445512505, 2.37794142916099814416965707949, 2.69294332589138691533521372601, 2.97296350566869200410881877135, 3.66836286189505578103644705043, 4.30164505802317307492007249569, 4.73529025010744677752477421678, 4.74090596514049221835074435982, 5.52522631346523749459457208012, 5.60122303682821828430940295754, 5.96426032676046222619033011623, 6.70572607022142048531350673876, 6.94788494901748043621824269016, 7.07565868062094417145466418044, 8.024423368007723938671909323732, 8.049278148922435234064451192165, 8.220724802447683652948001314610, 8.455247075133481006299115217412

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