↑

Properties

Label 4-2592e2-1.1-c1e2-0-36
Degree $4$
Conductor $6718464$
Sign $1$
Analytic cond. $428.375$
Root an. cond. $4.54942$
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  − 2·5-s − 2·7-s + 2·11-s + 2·13-s − 8·19-s + 6·23-s − 7·25-s + 10·29-s − 10·31-s + 4·35-s + 8·37-s − 14·41-s − 10·43-s − 2·47-s − 5·49-s + 8·53-s − 4·55-s − 14·59-s − 6·61-s − 4·65-s − 10·67-s + 4·71-s − 4·77-s − 22·79-s + 6·83-s − 16·89-s − 4·91-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.755·7-s + 0.603·11-s + 0.554·13-s − 1.83·19-s + 1.25·23-s − 7/5·25-s + 1.85·29-s − 1.79·31-s + 0.676·35-s + 1.31·37-s − 2.18·41-s − 1.52·43-s − 0.291·47-s − 5/7·49-s + 1.09·53-s − 0.539·55-s − 1.82·59-s − 0.768·61-s − 0.496·65-s − 1.22·67-s + 0.474·71-s − 0.455·77-s − 2.47·79-s + 0.658·83-s − 1.69·89-s − 0.419·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6718464\)    =    \(2^{10} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(428.375\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 6718464,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
good5$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.5.c_l
7$D_{4}$ \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_j
11$D_{4}$ \( 1 - 2 T + 17 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_r
13$D_{4}$ \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_d
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.17.a_k
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$D_{4}$ \( 1 - 6 T + 49 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_bx
29$D_{4}$ \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.29.ak_ch
31$D_{4}$ \( 1 + 10 T + 81 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_dd
37$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_co
41$D_{4}$ \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.41.o_ed
43$C_2^2$ \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_cf
47$D_{4}$ \( 1 + 2 T + 41 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_bp
53$D_{4}$ \( 1 - 8 T + 98 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_du
59$D_{4}$ \( 1 + 14 T + 113 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.59.o_ej
61$D_{4}$ \( 1 + 6 T + 107 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_ed
67$D_{4}$ \( 1 + 10 T + 105 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.67.k_eb
71$D_{4}$ \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.71.ae_by
73$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \) 2.73.a_es
79$D_{4}$ \( 1 + 22 T + 273 T^{2} + 22 p T^{3} + p^{2} T^{4} \) 2.79.w_kn
83$D_{4}$ \( 1 - 6 T + 169 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_gn
89$D_{4}$ \( 1 + 16 T + 218 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.89.q_ik
97$D_{4}$ \( 1 - 2 T + 171 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_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

−8.633423619948055414646113075618, −8.362426917531042672730256352356, −7.922721402596506256821964482028, −7.70714549939815560863065586906, −6.97762241430200115177365292856, −6.75600373080830338541019597129, −6.52281169978247837164571409321, −6.11350928907050060360986926898, −5.61976924248753063720234790252, −5.16896438118169749556681844709, −4.50747141599613733417031047079, −4.34722039498800434535699851600, −3.80543944869264623182005517821, −3.49843136992129748803582935263, −2.97192937483187741065428773673, −2.56820421880992707957959637341, −1.60347169177601591041401800268, −1.42073272012521395322399751058, 0, 0, 1.42073272012521395322399751058, 1.60347169177601591041401800268, 2.56820421880992707957959637341, 2.97192937483187741065428773673, 3.49843136992129748803582935263, 3.80543944869264623182005517821, 4.34722039498800434535699851600, 4.50747141599613733417031047079, 5.16896438118169749556681844709, 5.61976924248753063720234790252, 6.11350928907050060360986926898, 6.52281169978247837164571409321, 6.75600373080830338541019597129, 6.97762241430200115177365292856, 7.70714549939815560863065586906, 7.922721402596506256821964482028, 8.362426917531042672730256352356, 8.633423619948055414646113075618

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