↑

Properties

Label 4-435e2-1.1-c1e2-0-10
Degree $4$
Conductor $189225$
Sign $1$
Analytic cond. $12.0651$
Root an. cond. $1.86373$
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 + 2·3-s + 4-s + 2·5-s + 2·6-s + 2·7-s + 3·8-s + 3·9-s + 2·10-s − 7·11-s + 2·12-s − 4·13-s + 2·14-s + 4·15-s + 16-s − 6·17-s + 3·18-s + 2·19-s + 2·20-s + 4·21-s − 7·22-s + 9·23-s + 6·24-s + 3·25-s − 4·26-s + 4·27-s + 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s + 0.816·6-s + 0.755·7-s + 1.06·8-s + 9-s + 0.632·10-s − 2.11·11-s + 0.577·12-s − 1.10·13-s + 0.534·14-s + 1.03·15-s + 1/4·16-s − 1.45·17-s + 0.707·18-s + 0.458·19-s + 0.447·20-s + 0.872·21-s − 1.49·22-s + 1.87·23-s + 1.22·24-s + 3/5·25-s − 0.784·26-s + 0.769·27-s + 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(189225\)    =    \(3^{2} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(12.0651\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 189225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.772433380\)
\(L(\frac12)\) \(\approx\) \(4.772433380\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
29$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \) 2.2.ab_a
7$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_ac
11$D_{4}$ \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.11.h_be
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$D_{4}$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_ba
19$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_w
23$D_{4}$ \( 1 - 9 T + 62 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.23.aj_ck
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$D_{4}$ \( 1 - 9 T + 56 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.37.aj_ce
41$D_{4}$ \( 1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_cm
43$D_{4}$ \( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.43.ad_by
47$D_{4}$ \( 1 + 18 T + 158 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.47.s_gc
53$D_{4}$ \( 1 + 11 T + 132 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.53.l_fc
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$D_{4}$ \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.61.ak_fa
67$D_{4}$ \( 1 - 2 T - 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_as
71$D_{4}$ \( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.71.o_gs
73$D_{4}$ \( 1 + 9 T + 128 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.73.j_ey
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.79.y_lq
83$D_{4}$ \( 1 + T + 162 T^{2} + p T^{3} + p^{2} T^{4} \) 2.83.b_gg
89$C_4$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.89.ai_ew
97$D_{4}$ \( 1 - T + 156 T^{2} - p T^{3} + p^{2} T^{4} \) 2.97.ab_ga
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

−11.36251397374127041343089626201, −10.96417206488600490263655285433, −10.34264019006548469411454494862, −10.10904995326953672334215539315, −9.648021302574650797261891232320, −9.118052207317669976865804380000, −8.543204798214796364896863540620, −8.112249447851318817285081961621, −7.75235415881385980632209288744, −7.08278780101326358501288651886, −7.00095725827243836204529869390, −6.19888645960561360340900749760, −5.29906685322338603656687617056, −5.11028811136306908507362470100, −4.54100118367316077581297147597, −4.28680635450717115076616591496, −2.96591010180264531646197447851, −2.50140308063169562697748210635, −2.46123361879166738208870635262, −1.36360332662919032380159427521, 1.36360332662919032380159427521, 2.46123361879166738208870635262, 2.50140308063169562697748210635, 2.96591010180264531646197447851, 4.28680635450717115076616591496, 4.54100118367316077581297147597, 5.11028811136306908507362470100, 5.29906685322338603656687617056, 6.19888645960561360340900749760, 7.00095725827243836204529869390, 7.08278780101326358501288651886, 7.75235415881385980632209288744, 8.112249447851318817285081961621, 8.543204798214796364896863540620, 9.118052207317669976865804380000, 9.648021302574650797261891232320, 10.10904995326953672334215539315, 10.34264019006548469411454494862, 10.96417206488600490263655285433, 11.36251397374127041343089626201

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