Properties

Label 4-2070e2-1.1-c1e2-0-2
Degree $4$
Conductor $4284900$
Sign $1$
Analytic cond. $273.208$
Root an. cond. $4.06559$
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  + 2·2-s + 3·4-s − 2·5-s + 3·7-s + 4·8-s − 4·10-s + 7·11-s + 3·13-s + 6·14-s + 5·16-s − 3·17-s + 19-s − 6·20-s + 14·22-s + 2·23-s + 3·25-s + 6·26-s + 9·28-s − 2·29-s − 5·31-s + 6·32-s − 6·34-s − 6·35-s + 16·37-s + 2·38-s − 8·40-s + 9·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.894·5-s + 1.13·7-s + 1.41·8-s − 1.26·10-s + 2.11·11-s + 0.832·13-s + 1.60·14-s + 5/4·16-s − 0.727·17-s + 0.229·19-s − 1.34·20-s + 2.98·22-s + 0.417·23-s + 3/5·25-s + 1.17·26-s + 1.70·28-s − 0.371·29-s − 0.898·31-s + 1.06·32-s − 1.02·34-s − 1.01·35-s + 2.63·37-s + 0.324·38-s − 1.26·40-s + 1.40·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4284900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4284900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(273.208\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2070} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4284900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(9.042783621\)
\(L(\frac12)\) \(\approx\) \(9.042783621\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 7 T + 31 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 5 T + 47 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 29 T + 349 T^{2} - 29 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 9 T + 211 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
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

−9.228192260236810906156805897813, −8.852819216778846306919084780360, −8.395717147071233166762378179916, −8.191381251989494102129154696399, −7.51337932851913533906279946267, −7.41609433511581774568989715472, −6.81682431454569900359642108095, −6.52754052796858718914512777567, −6.11311939324029171015771297191, −5.81021230889943320094525581531, −4.99000246623524088649143621828, −4.97698543825620063644638945037, −4.25771499872425734760953895160, −4.08996027210217114618458707711, −3.56612978338512225674629872144, −3.51116061532589262500833595437, −2.36505019626353441922533065140, −2.25339301567425683445484211469, −1.17981764933870209034850387785, −1.06854396603220431591782468674, 1.06854396603220431591782468674, 1.17981764933870209034850387785, 2.25339301567425683445484211469, 2.36505019626353441922533065140, 3.51116061532589262500833595437, 3.56612978338512225674629872144, 4.08996027210217114618458707711, 4.25771499872425734760953895160, 4.97698543825620063644638945037, 4.99000246623524088649143621828, 5.81021230889943320094525581531, 6.11311939324029171015771297191, 6.52754052796858718914512777567, 6.81682431454569900359642108095, 7.41609433511581774568989715472, 7.51337932851913533906279946267, 8.191381251989494102129154696399, 8.395717147071233166762378179916, 8.852819216778846306919084780360, 9.228192260236810906156805897813

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