Properties

Label 4-1520e2-1.1-c1e2-0-30
Degree $4$
Conductor $2310400$
Sign $1$
Analytic cond. $147.313$
Root an. cond. $3.48385$
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  − 2·3-s + 2·5-s − 4·7-s − 2·13-s − 4·15-s − 2·19-s + 8·21-s + 3·25-s + 2·27-s − 4·31-s − 8·35-s + 10·37-s + 4·39-s − 12·41-s − 4·43-s − 12·47-s − 2·49-s − 18·53-s + 4·57-s + 4·61-s − 4·65-s − 10·67-s + 12·71-s − 8·73-s − 6·75-s + 8·79-s − 81-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 1.51·7-s − 0.554·13-s − 1.03·15-s − 0.458·19-s + 1.74·21-s + 3/5·25-s + 0.384·27-s − 0.718·31-s − 1.35·35-s + 1.64·37-s + 0.640·39-s − 1.87·41-s − 0.609·43-s − 1.75·47-s − 2/7·49-s − 2.47·53-s + 0.529·57-s + 0.512·61-s − 0.496·65-s − 1.22·67-s + 1.42·71-s − 0.936·73-s − 0.692·75-s + 0.900·79-s − 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2310400\)    =    \(2^{8} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(147.313\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2310400,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_e
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$D_{4}$ \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_y
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.17.a_w
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.23.a_bi
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.29.a_bu
31$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_cc
37$D_{4}$ \( 1 - 10 T + 96 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.37.ak_ds
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_bq
47$D_{4}$ \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_de
53$D_{4}$ \( 1 + 18 T + 184 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.53.s_hc
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.59.a_cs
61$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_s
67$D_{4}$ \( 1 + 10 T + 156 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.67.k_ga
71$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.71.am_gk
73$D_{4}$ \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_fu
79$D_{4}$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.79.ai_ew
83$C_2^2$ \( 1 + 154 T^{2} + p^{2} T^{4} \) 2.83.a_fy
89$D_{4}$ \( 1 + 24 T + 310 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.89.y_ly
97$D_{4}$ \( 1 + 2 T - 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_abw
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

−9.473626759687787217798698177772, −9.131492239069850516877154401619, −8.300402196192609640328527693249, −8.236698311914008664717230394914, −7.65297966437495532642683286587, −6.86660222715208306520891890350, −6.65579464459071028814093729249, −6.52911533516102420844842400796, −5.90671211183201936218083933415, −5.77062955146184680594812156109, −5.08084977293120133769417440291, −4.96107038559969524165743387047, −4.27724304767652366185192285777, −3.63556900134341021233384687215, −2.96373216224476934882459709097, −2.88404529689345872012166958279, −1.92361700076206183489253495483, −1.38994657014257316200534284835, 0, 0, 1.38994657014257316200534284835, 1.92361700076206183489253495483, 2.88404529689345872012166958279, 2.96373216224476934882459709097, 3.63556900134341021233384687215, 4.27724304767652366185192285777, 4.96107038559969524165743387047, 5.08084977293120133769417440291, 5.77062955146184680594812156109, 5.90671211183201936218083933415, 6.52911533516102420844842400796, 6.65579464459071028814093729249, 6.86660222715208306520891890350, 7.65297966437495532642683286587, 8.236698311914008664717230394914, 8.300402196192609640328527693249, 9.131492239069850516877154401619, 9.473626759687787217798698177772

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