↑

Properties

Label 4-930e2-1.1-c1e2-0-2
Degree $4$
Conductor $864900$
Sign $1$
Analytic cond. $55.1467$
Root an. cond. $2.72508$
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-s + 3·4-s − 5-s − 2·6-s − 5·7-s + 4·8-s − 2·10-s + 11-s − 3·12-s − 4·13-s − 10·14-s + 15-s + 5·16-s − 4·17-s − 4·19-s − 3·20-s + 5·21-s + 2·22-s + 4·23-s − 4·24-s − 8·26-s + 27-s − 15·28-s − 18·29-s + 2·30-s − 11·31-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.447·5-s − 0.816·6-s − 1.88·7-s + 1.41·8-s − 0.632·10-s + 0.301·11-s − 0.866·12-s − 1.10·13-s − 2.67·14-s + 0.258·15-s + 5/4·16-s − 0.970·17-s − 0.917·19-s − 0.670·20-s + 1.09·21-s + 0.426·22-s + 0.834·23-s − 0.816·24-s − 1.56·26-s + 0.192·27-s − 2.83·28-s − 3.34·29-s + 0.365·30-s − 1.97·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(864900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(55.1467\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 864900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.171845662\)
\(L(\frac12)\) \(\approx\) \(1.171845662\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
31$C_2$ \( 1 + 11 T + p T^{2} \)
good7$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.f_s
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) 2.11.ab_ak
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_d
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_ab
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_ad
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.23.ae_by
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.29.s_fj
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_abh
41$C_2^2$ \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.41.ac_abl
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.i_v
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.53.aj_bc
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.59.ad_aby
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2^2$ \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.67.ak_bh
71$C_2^2$ \( 1 - 4 T - 55 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.71.ae_acd
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_acr
79$C_2^2$ \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_cn
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.83.aj_ac
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.97.c_hn
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

−10.44059175555148158650831935479, −9.857504565905466463292229302038, −9.564701528093817623249267999424, −8.990929058070490281775503587306, −8.886089635943010266331329871292, −7.933777377430778476396034650552, −7.43737736408117011508053811949, −7.04511256942174932746265243249, −6.70767701792017356476275563227, −6.58137146291907445886427305690, −5.71234934582740554246669523891, −5.43557313652231633360915989398, −5.31462483362516590380693252744, −4.20071517198686891558781611059, −4.16317205055884609459573648081, −3.49955072810898198055494204159, −3.23778548332665826473118592906, −2.18595838288899145473078484774, −2.13496396510462275707293512527, −0.38548791886378029315999885008, 0.38548791886378029315999885008, 2.13496396510462275707293512527, 2.18595838288899145473078484774, 3.23778548332665826473118592906, 3.49955072810898198055494204159, 4.16317205055884609459573648081, 4.20071517198686891558781611059, 5.31462483362516590380693252744, 5.43557313652231633360915989398, 5.71234934582740554246669523891, 6.58137146291907445886427305690, 6.70767701792017356476275563227, 7.04511256942174932746265243249, 7.43737736408117011508053811949, 7.933777377430778476396034650552, 8.886089635943010266331329871292, 8.990929058070490281775503587306, 9.564701528093817623249267999424, 9.857504565905466463292229302038, 10.44059175555148158650831935479

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