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Properties

Label 4-66e4-1.1-c1e2-0-12
Degree $4$
Conductor $18974736$
Sign $1$
Analytic cond. $1209.84$
Root an. cond. $5.89769$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s + 7·13-s − 7·17-s + 19-s − 4·23-s − 8·25-s − 15·29-s − 5·31-s − 35-s + 3·37-s − 15·41-s − 5·47-s − 2·49-s − 3·53-s + 9·59-s + 7·61-s + 7·65-s + 8·67-s + 15·71-s − 13·73-s − 21·79-s + 3·83-s − 7·85-s + 8·89-s − 7·91-s + 95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 1.94·13-s − 1.69·17-s + 0.229·19-s − 0.834·23-s − 8/5·25-s − 2.78·29-s − 0.898·31-s − 0.169·35-s + 0.493·37-s − 2.34·41-s − 0.729·47-s − 2/7·49-s − 0.412·53-s + 1.17·59-s + 0.896·61-s + 0.868·65-s + 0.977·67-s + 1.78·71-s − 1.52·73-s − 2.36·79-s + 0.329·83-s − 0.759·85-s + 0.847·89-s − 0.733·91-s + 0.102·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18974736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18974736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(18974736\)    =    \(2^{4} \cdot 3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1209.84\)
Root analytic conductor: \(5.89769\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 18974736,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5$D_{4}$ \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_j
7$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) 2.7.b_d
13$D_{4}$ \( 1 - 7 T + 37 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.13.ah_bl
17$D_{4}$ \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.17.h_bt
19$D_{4}$ \( 1 - T + 27 T^{2} - p T^{3} + p^{2} T^{4} \) 2.19.ab_bb
23$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_be
29$D_{4}$ \( 1 + 15 T + 113 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.29.p_ej
31$D_{4}$ \( 1 + 5 T + 57 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.31.f_cf
37$D_{4}$ \( 1 - 3 T + 65 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.37.ad_cn
41$D_{4}$ \( 1 + 15 T + 137 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.41.p_fh
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$D_{4}$ \( 1 + 5 T + 99 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.47.f_dv
53$D_{4}$ \( 1 + 3 T + 77 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_cz
59$D_{4}$ \( 1 - 9 T + 107 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.59.aj_ed
61$D_{4}$ \( 1 - 7 T + 133 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.61.ah_fd
67$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_cs
71$D_{4}$ \( 1 - 15 T + 197 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.71.ap_hp
73$D_{4}$ \( 1 + 13 T + 157 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.73.n_gb
79$D_{4}$ \( 1 + 21 T + 257 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.79.v_jx
83$D_{4}$ \( 1 - 3 T - 43 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.83.ad_abr
89$D_{4}$ \( 1 - 8 T + 174 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.89.ai_gs
97$D_{4}$ \( 1 + 9 T + 113 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.97.j_ej
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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

−8.082978924517888709082999782039, −8.061752574979440662322928413764, −7.44220594515300320627774459734, −6.97882899379567160572437508545, −6.66229631407951040189487139250, −6.44944907011428744663535405645, −5.86917946008325192238873979544, −5.74519379358932308061752559075, −5.32684827605482305094910056943, −4.97593458879190908017176196144, −4.10498579378014881980705995759, −4.00882133342716792405913109988, −3.65356178572082789065051524930, −3.37908160669909340816606638970, −2.50914578970101574220987541427, −2.18086032811438569089565798527, −1.58992163868220489787068051116, −1.43342325611455506712762512021, 0, 0, 1.43342325611455506712762512021, 1.58992163868220489787068051116, 2.18086032811438569089565798527, 2.50914578970101574220987541427, 3.37908160669909340816606638970, 3.65356178572082789065051524930, 4.00882133342716792405913109988, 4.10498579378014881980705995759, 4.97593458879190908017176196144, 5.32684827605482305094910056943, 5.74519379358932308061752559075, 5.86917946008325192238873979544, 6.44944907011428744663535405645, 6.66229631407951040189487139250, 6.97882899379567160572437508545, 7.44220594515300320627774459734, 8.061752574979440662322928413764, 8.082978924517888709082999782039

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