Properties

Label 4-1386e2-1.1-c1e2-0-24
Degree $4$
Conductor $1920996$
Sign $1$
Analytic cond. $122.484$
Root an. cond. $3.32675$
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  − 2-s + 4·7-s + 8-s − 11-s + 8·13-s − 4·14-s − 16-s − 17-s + 3·19-s + 22-s − 23-s + 5·25-s − 8·26-s + 2·29-s − 6·31-s + 34-s + 3·37-s − 3·38-s + 12·41-s + 2·43-s + 46-s − 47-s + 9·49-s − 5·50-s + 4·56-s − 2·58-s + 7·59-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.51·7-s + 0.353·8-s − 0.301·11-s + 2.21·13-s − 1.06·14-s − 1/4·16-s − 0.242·17-s + 0.688·19-s + 0.213·22-s − 0.208·23-s + 25-s − 1.56·26-s + 0.371·29-s − 1.07·31-s + 0.171·34-s + 0.493·37-s − 0.486·38-s + 1.87·41-s + 0.304·43-s + 0.147·46-s − 0.145·47-s + 9/7·49-s − 0.707·50-s + 0.534·56-s − 0.262·58-s + 0.911·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1920996\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(122.484\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1386} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1920996,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.408500057\)
\(L(\frac12)\) \(\approx\) \(2.408500057\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
11$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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

−9.565120466471218618658680802780, −9.236724046381275993647095006301, −9.002338443464102858704753666565, −8.545816587868687200787233232187, −8.104830595294451676100375430107, −8.029681256908266097291219361674, −7.44181061763764898260085349480, −7.18304020247927490510410013078, −6.35513047013373148306125061944, −6.23161248907840701119520015073, −5.63451907964691722057356383326, −5.06435612130953063479283479425, −4.88413782263441815320906466744, −4.19118632286208238114467399023, −3.74910637044342925611170918311, −3.34365604398416506472396542466, −2.40532938657690266230165235585, −2.00178093046359585047280429132, −1.07225671076757514254252704793, −0.961111099051834868054928552456, 0.961111099051834868054928552456, 1.07225671076757514254252704793, 2.00178093046359585047280429132, 2.40532938657690266230165235585, 3.34365604398416506472396542466, 3.74910637044342925611170918311, 4.19118632286208238114467399023, 4.88413782263441815320906466744, 5.06435612130953063479283479425, 5.63451907964691722057356383326, 6.23161248907840701119520015073, 6.35513047013373148306125061944, 7.18304020247927490510410013078, 7.44181061763764898260085349480, 8.029681256908266097291219361674, 8.104830595294451676100375430107, 8.545816587868687200787233232187, 9.002338443464102858704753666565, 9.236724046381275993647095006301, 9.565120466471218618658680802780

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