↑

Properties

Label 4-1150e2-1.1-c1e2-0-9
Degree $4$
Conductor $1322500$
Sign $1$
Analytic cond. $84.3237$
Root an. cond. $3.03031$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 3·4-s − 6·6-s − 3·7-s + 4·8-s + 4·9-s − 7·11-s − 9·12-s − 3·13-s − 6·14-s + 5·16-s − 3·17-s + 8·18-s + 19-s + 9·21-s − 14·22-s + 2·23-s − 12·24-s − 6·26-s − 6·27-s − 9·28-s + 2·29-s − 5·31-s + 6·32-s + 21·33-s − 6·34-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.73·3-s + 3/2·4-s − 2.44·6-s − 1.13·7-s + 1.41·8-s + 4/3·9-s − 2.11·11-s − 2.59·12-s − 0.832·13-s − 1.60·14-s + 5/4·16-s − 0.727·17-s + 1.88·18-s + 0.229·19-s + 1.96·21-s − 2.98·22-s + 0.417·23-s − 2.44·24-s − 1.17·26-s − 1.15·27-s − 1.70·28-s + 0.371·29-s − 0.898·31-s + 1.06·32-s + 3.65·33-s − 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1322500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1322500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1322500\)    =    \(2^{2} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(84.3237\)
Root analytic conductor: \(3.03031\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1322500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
23$C_1$ \( ( 1 - T )^{2} \)
good3$C_4$ \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.3.d_f
7$D_{4}$ \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_n
11$D_{4}$ \( 1 + 7 T + 31 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.11.h_bf
13$D_{4}$ \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.13.d_z
17$D_{4}$ \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_h
19$D_{4}$ \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) 2.19.ab_j
29$D_{4}$ \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.29.ac_bu
31$D_{4}$ \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.31.f_bn
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
41$D_{4}$ \( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_cv
43$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_bm
47$D_{4}$ \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_de
53$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_cs
59$D_{4}$ \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.59.o_fy
61$D_{4}$ \( 1 - 5 T + 47 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.61.af_bv
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$D_{4}$ \( 1 + 29 T + 349 T^{2} + 29 p T^{3} + p^{2} T^{4} \) 2.71.bd_nl
73$D_{4}$ \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.73.k_cc
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$D_{4}$ \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.83.i_fa
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$D_{4}$ \( 1 + 9 T + 211 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.97.j_id
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

−10.01012643253479951845004145681, −9.340803057304897498554607170347, −8.734534460783700156842579658814, −8.318454964036439308329543288967, −7.42196309193802929658620164370, −7.40895269131913922781407732370, −6.93458094345463377609827273747, −6.54621121653590608775749963017, −5.90074627114363158138530160065, −5.81034043323570385534607923940, −5.24746536813923552644599284364, −4.99481539553635292729368058160, −4.70872339687791640841690438036, −4.00002778728647914320899627752, −3.25216576786624909802711632427, −3.04004680587820482871752922772, −2.32756097422350121058519862216, −1.65439286162488817182646439480, 0, 0, 1.65439286162488817182646439480, 2.32756097422350121058519862216, 3.04004680587820482871752922772, 3.25216576786624909802711632427, 4.00002778728647914320899627752, 4.70872339687791640841690438036, 4.99481539553635292729368058160, 5.24746536813923552644599284364, 5.81034043323570385534607923940, 5.90074627114363158138530160065, 6.54621121653590608775749963017, 6.93458094345463377609827273747, 7.40895269131913922781407732370, 7.42196309193802929658620164370, 8.318454964036439308329543288967, 8.734534460783700156842579658814, 9.340803057304897498554607170347, 10.01012643253479951845004145681

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