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Properties

Label 4-733e2-1.1-c1e2-0-3
Degree $4$
Conductor $537289$
Sign $1$
Analytic cond. $34.2580$
Root an. cond. $2.41930$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4-s − 6·7-s + 3·9-s − 2·12-s − 3·16-s − 3·17-s − 4·19-s + 12·21-s − 12·23-s + 7·25-s − 10·27-s − 6·28-s + 4·31-s + 3·36-s + 37-s − 6·41-s − 18·43-s − 12·47-s + 6·48-s + 17·49-s + 6·51-s − 3·53-s + 8·57-s − 6·59-s + 7·61-s − 18·63-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s − 2.26·7-s + 9-s − 0.577·12-s − 3/4·16-s − 0.727·17-s − 0.917·19-s + 2.61·21-s − 2.50·23-s + 7/5·25-s − 1.92·27-s − 1.13·28-s + 0.718·31-s + 1/2·36-s + 0.164·37-s − 0.937·41-s − 2.74·43-s − 1.75·47-s + 0.866·48-s + 17/7·49-s + 0.840·51-s − 0.412·53-s + 1.05·57-s − 0.781·59-s + 0.896·61-s − 2.26·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 537289 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 537289 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(537289\)    =    \(733^{2}\)
Sign: $1$
Analytic conductor: \(34.2580\)
Root analytic conductor: \(2.41930\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 537289,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad733$C_2$ \( 1 + 50 T + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.2.a_ab
3$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_b
5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.5.a_ah
7$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.g_t
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.13.a_n
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2^2$ \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.23.m_ct
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ae_ap
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.ab_abk
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_af
43$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.s_fv
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.47.m_fa
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_abs
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_ax
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.61.ah_am
67$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.67.a_adi
71$C_2^2$ \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_df
73$C_2^2$ \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.73.o_et
79$C_2^2$ \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.79.aq_gv
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_cj
89$C_2^2$ \( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.89.j_ai
97$C_2^2$ \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.97.n_cu
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.14164276421527542914407547445, −10.01959245468582274409750616730, −9.519511829994809298905453513237, −8.985357687180411852310610857588, −8.444467231592149303343311819409, −8.079297352509640607070769342372, −7.18363327465562344855308625499, −6.99927751879199718178472963701, −6.41468032191670716898618560490, −6.21552850004090180794858495756, −6.15377451087112072461403958677, −5.28901166069917357520511443186, −4.64066617713573144018594578558, −4.29650927196788135828603443371, −3.45809855715132784665238421942, −3.21909493112272276546650029755, −2.26345029351565778769938561634, −1.73007936420359588716063737553, 0, 0, 1.73007936420359588716063737553, 2.26345029351565778769938561634, 3.21909493112272276546650029755, 3.45809855715132784665238421942, 4.29650927196788135828603443371, 4.64066617713573144018594578558, 5.28901166069917357520511443186, 6.15377451087112072461403958677, 6.21552850004090180794858495756, 6.41468032191670716898618560490, 6.99927751879199718178472963701, 7.18363327465562344855308625499, 8.079297352509640607070769342372, 8.444467231592149303343311819409, 8.985357687180411852310610857588, 9.519511829994809298905453513237, 10.01959245468582274409750616730, 10.14164276421527542914407547445

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