↑

Properties

Label 4-450e2-1.1-c1e2-0-9
Degree $4$
Conductor $202500$
Sign $1$
Analytic cond. $12.9115$
Root an. cond. $1.89559$
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 + 3·3-s + 3·6-s − 7-s − 8-s + 6·9-s + 2·11-s − 6·13-s − 14-s − 16-s + 4·17-s + 6·18-s + 12·19-s − 3·21-s + 2·22-s + 23-s − 3·24-s − 6·26-s + 9·27-s − 9·29-s + 2·31-s + 6·33-s + 4·34-s − 4·37-s + 12·38-s − 18·39-s + 11·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s + 0.603·11-s − 1.66·13-s − 0.267·14-s − 1/4·16-s + 0.970·17-s + 1.41·18-s + 2.75·19-s − 0.654·21-s + 0.426·22-s + 0.208·23-s − 0.612·24-s − 1.17·26-s + 1.73·27-s − 1.67·29-s + 0.359·31-s + 1.04·33-s + 0.685·34-s − 0.657·37-s + 1.94·38-s − 2.88·39-s + 1.71·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(202500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(12.9115\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 202500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.363691276\)
\(L(\frac12)\) \(\approx\) \(4.363691276\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 - p T + p T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.b_ag
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_ah
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_x
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.19.am_cw
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) 2.23.ab_aw
29$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.29.j_ca
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_abb
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2^2$ \( 1 - 11 T + 80 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.41.al_dc
43$C_2^2$ \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_abb
47$C_2^2$ \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.47.h_c
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_abr
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.61.ah_am
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.l_cc
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.73.ai_gg
79$C_2^2$ \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_cn
83$C_2^2$ \( 1 + 11 T + 38 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.83.l_bm
89$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.89.ac_gx
97$C_2^2$ \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_abh
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

−11.48966756677868233448556566790, −10.89494979573783573752794986184, −10.01017730899387037747275254222, −9.801225634631384391720901245147, −9.550844405896477565828687250390, −9.236182684772098707220787825060, −8.648292995987210437224674686512, −8.097339354425975875850425490863, −7.44251812564754360861457906003, −7.33363924584719492375648275269, −7.04986705036110303408472636946, −6.01346710259467282913268356923, −5.52079711748507274723474061707, −5.01338013227043856912891670704, −4.43777152634048219273672562745, −3.66955792786175549294448284817, −3.37561479560052560478036735673, −2.87865039617803272921413129887, −2.21546056966703936744975929825, −1.21655190000704858499829476092, 1.21655190000704858499829476092, 2.21546056966703936744975929825, 2.87865039617803272921413129887, 3.37561479560052560478036735673, 3.66955792786175549294448284817, 4.43777152634048219273672562745, 5.01338013227043856912891670704, 5.52079711748507274723474061707, 6.01346710259467282913268356923, 7.04986705036110303408472636946, 7.33363924584719492375648275269, 7.44251812564754360861457906003, 8.097339354425975875850425490863, 8.648292995987210437224674686512, 9.236182684772098707220787825060, 9.550844405896477565828687250390, 9.801225634631384391720901245147, 10.01017730899387037747275254222, 10.89494979573783573752794986184, 11.48966756677868233448556566790

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