↑

Properties

Label 4-3024e2-1.1-c1e2-0-17
Degree $4$
Conductor $9144576$
Sign $1$
Analytic cond. $583.066$
Root an. cond. $4.91393$
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  − 7-s + 6·11-s − 3·13-s − 3·17-s − 5·19-s − 6·23-s − 5·25-s + 3·29-s + 2·31-s − 7·37-s + 3·41-s + 3·43-s + 18·47-s − 6·49-s − 9·53-s + 30·59-s + 3·73-s − 6·77-s + 9·83-s − 3·89-s + 3·91-s − 3·97-s + 24·101-s + 8·103-s + 3·107-s − 11·109-s − 9·113-s + ⋯
L(s)  = 1  − 0.377·7-s + 1.80·11-s − 0.832·13-s − 0.727·17-s − 1.14·19-s − 1.25·23-s − 25-s + 0.557·29-s + 0.359·31-s − 1.15·37-s + 0.468·41-s + 0.457·43-s + 2.62·47-s − 6/7·49-s − 1.23·53-s + 3.90·59-s + 0.351·73-s − 0.683·77-s + 0.987·83-s − 0.317·89-s + 0.314·91-s − 0.304·97-s + 2.38·101-s + 0.788·103-s + 0.290·107-s − 1.05·109-s − 0.846·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(583.066\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.678614803\)
\(L(\frac12)\) \(\approx\) \(1.678614803\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.5.a_f
11$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_x
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.d_q
17$C_2^2$ \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_u
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.19.f_g
23$C_2^2$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_bj
29$C_2^2$ \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_au
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.31.ac_cl
37$C_2^2$ \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.37.h_m
41$C_2^2$ \( 1 - 3 T + 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_bs
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.ad_bu
47$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.47.as_gt
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.53.j_bc
59$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.59.abe_nf
61$C_2^2$ \( 1 - 119 T^{2} + p^{2} T^{4} \) 2.61.a_aep
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.a_ef
71$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.71.a_abi
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.ad_cy
79$C_2^2$ \( 1 - 155 T^{2} + p^{2} T^{4} \) 2.79.a_afz
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.83.aj_ac
89$C_2^2$ \( 1 + 3 T + 92 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.89.d_do
97$C_2^2$ \( 1 + 3 T + 100 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.97.d_dw
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

−8.863303641710569400758869881855, −8.766564795737429063828239305259, −8.072806085652691647947362662841, −7.927817999045557620103650863461, −7.30521783709530839709076233291, −6.91250975668837177835250841705, −6.72140558858466756734405577353, −6.29202745347741459106146260346, −5.93355277986632314330356637683, −5.64353420449175431426600555651, −4.93375262417516206642840198396, −4.61268337361711598924353731833, −4.02081157878349148111615025652, −3.90491175832975606772390203475, −3.55682627541920020372288068575, −2.70504336890518614108390483869, −2.15660099687245580252773285461, −2.05660924737912003910928759973, −1.16337622353428264424480523494, −0.43129282618097889744324184456, 0.43129282618097889744324184456, 1.16337622353428264424480523494, 2.05660924737912003910928759973, 2.15660099687245580252773285461, 2.70504336890518614108390483869, 3.55682627541920020372288068575, 3.90491175832975606772390203475, 4.02081157878349148111615025652, 4.61268337361711598924353731833, 4.93375262417516206642840198396, 5.64353420449175431426600555651, 5.93355277986632314330356637683, 6.29202745347741459106146260346, 6.72140558858466756734405577353, 6.91250975668837177835250841705, 7.30521783709530839709076233291, 7.927817999045557620103650863461, 8.072806085652691647947362662841, 8.766564795737429063828239305259, 8.863303641710569400758869881855

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