Properties

Label 4-968e2-1.1-c1e2-0-18
Degree $4$
Conductor $937024$
Sign $1$
Analytic cond. $59.7454$
Root an. cond. $2.78020$
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 − 4·5-s − 2·7-s − 2·13-s − 8·15-s − 8·17-s − 10·19-s − 4·21-s + 6·23-s + 5·25-s − 2·27-s + 2·29-s − 6·31-s + 8·35-s + 8·37-s − 4·39-s − 12·41-s − 16·43-s − 10·47-s − 8·49-s − 16·51-s − 8·53-s − 20·57-s + 20·59-s − 20·61-s + 8·65-s + 10·67-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s − 0.755·7-s − 0.554·13-s − 2.06·15-s − 1.94·17-s − 2.29·19-s − 0.872·21-s + 1.25·23-s + 25-s − 0.384·27-s + 0.371·29-s − 1.07·31-s + 1.35·35-s + 1.31·37-s − 0.640·39-s − 1.87·41-s − 2.43·43-s − 1.45·47-s − 8/7·49-s − 2.24·51-s − 1.09·53-s − 2.64·57-s + 2.60·59-s − 2.56·61-s + 0.992·65-s + 1.22·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 937024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 937024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(937024\)    =    \(2^{6} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(59.7454\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 937024,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
11 \( 1 \)
good3$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_e
5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_l
7$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_m
13$D_{4}$ \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_p
17$C_2^2$ \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bv
19$D_{4}$ \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.19.k_ci
23$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_bc
29$D_{4}$ \( 1 - 2 T + 47 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.29.ac_bv
31$D_{4}$ \( 1 + 6 T + 68 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_cq
37$D_{4}$ \( 1 - 8 T + 87 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_dj
41$D_{4}$ \( 1 + 12 T + 115 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_el
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$D_{4}$ \( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.47.k_em
53$D_{4}$ \( 1 + 8 T + 95 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_dr
59$D_{4}$ \( 1 - 20 T + 206 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.59.au_hy
61$D_{4}$ \( 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.61.u_ic
67$D_{4}$ \( 1 - 10 T + 132 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.67.ak_fc
71$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.71.am_gk
73$D_{4}$ \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.73.ae_dy
79$D_{4}$ \( 1 - 2 T - 84 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.79.ac_adg
83$D_{4}$ \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_fs
89$D_{4}$ \( 1 + 26 T + 335 T^{2} + 26 p T^{3} + p^{2} T^{4} \) 2.89.ba_mx
97$D_{4}$ \( 1 + 10 T + 171 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.97.k_gp
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

−9.516626269765392890927879657419, −9.501511968196858117506112342756, −8.650902924874522799302895792925, −8.556205353380160299294820233482, −8.157414227902775534504397459165, −8.095482075970109109116033868098, −7.24973641312428691407820782536, −6.80773166764290575951118455964, −6.62616174203198145342678498389, −6.27057469859998265642502470032, −5.20537907806558078833166978958, −4.79125939218276076433555481950, −4.35841078567303355082699433589, −3.90048880094687931693284327579, −3.24914766414677537668232638030, −3.15811078667144471847144663145, −2.31675742498248633477687592748, −1.85444800299435606854446538835, 0, 0, 1.85444800299435606854446538835, 2.31675742498248633477687592748, 3.15811078667144471847144663145, 3.24914766414677537668232638030, 3.90048880094687931693284327579, 4.35841078567303355082699433589, 4.79125939218276076433555481950, 5.20537907806558078833166978958, 6.27057469859998265642502470032, 6.62616174203198145342678498389, 6.80773166764290575951118455964, 7.24973641312428691407820782536, 8.095482075970109109116033868098, 8.157414227902775534504397459165, 8.556205353380160299294820233482, 8.650902924874522799302895792925, 9.501511968196858117506112342756, 9.516626269765392890927879657419

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