↑

Properties

Label 4-888e2-1.1-c1e2-0-10
Degree $4$
Conductor $788544$
Sign $1$
Analytic cond. $50.2782$
Root an. cond. $2.66283$
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 + 2·4-s + 4·5-s − 9-s − 8·10-s + 2·13-s − 4·16-s − 4·17-s + 2·18-s + 2·19-s + 8·20-s + 4·23-s + 8·25-s − 4·26-s + 4·29-s + 14·31-s + 8·32-s + 8·34-s − 2·36-s + 2·37-s − 4·38-s + 6·43-s − 4·45-s − 8·46-s + 10·49-s − 16·50-s + 4·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 1.78·5-s − 1/3·9-s − 2.52·10-s + 0.554·13-s − 16-s − 0.970·17-s + 0.471·18-s + 0.458·19-s + 1.78·20-s + 0.834·23-s + 8/5·25-s − 0.784·26-s + 0.742·29-s + 2.51·31-s + 1.41·32-s + 1.37·34-s − 1/3·36-s + 0.328·37-s − 0.648·38-s + 0.914·43-s − 0.596·45-s − 1.17·46-s + 10/7·49-s − 2.26·50-s + 0.554·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 788544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 788544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(788544\)    =    \(2^{6} \cdot 3^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(50.2782\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 788544,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.535331643\)
\(L(\frac12)\) \(\approx\) \(1.535331643\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
3$C_2$ \( 1 + T^{2} \)
37$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.5.ae_i
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.ac_c
17$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_i
19$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_c
23$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_i
29$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_i
31$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.31.ao_du
41$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.41.a_ada
43$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_s
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.47.a_ada
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.53.a_ady
59$C_2^2$ \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.59.q_ey
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.ac_c
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.79.ag_s
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 + p^{2} T^{4} \) 2.89.a_a
97$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.97.ag_s
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

−10.03614058697514720409731765198, −10.00894197084201249352971792845, −9.382959529560288784137842266253, −9.162412230702176245135928638978, −8.851201350627522612766167133662, −8.304589255539988244513612198993, −8.077792921521634501756229794256, −7.42443295238108130726202240504, −6.86026462605497141634042240493, −6.60194786051203801496840456412, −6.01554880926352772617924351487, −5.88308903033503156851089064996, −5.06414814738907386314366854437, −4.66785792895413612467476223045, −4.16027056685194571083056115565, −3.08282593212576704895997389526, −2.60693636046368350562204719383, −2.17697722052783151023676730091, −1.35496101087541295227087834142, −0.842860696805969197509408936226, 0.842860696805969197509408936226, 1.35496101087541295227087834142, 2.17697722052783151023676730091, 2.60693636046368350562204719383, 3.08282593212576704895997389526, 4.16027056685194571083056115565, 4.66785792895413612467476223045, 5.06414814738907386314366854437, 5.88308903033503156851089064996, 6.01554880926352772617924351487, 6.60194786051203801496840456412, 6.86026462605497141634042240493, 7.42443295238108130726202240504, 8.077792921521634501756229794256, 8.304589255539988244513612198993, 8.851201350627522612766167133662, 9.162412230702176245135928638978, 9.382959529560288784137842266253, 10.00894197084201249352971792845, 10.03614058697514720409731765198

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