↑

Properties

Label 4-2664e2-1.1-c1e2-0-11
Degree $4$
Conductor $7096896$
Sign $1$
Analytic cond. $452.504$
Root an. cond. $4.61217$
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  + 3·7-s − 3·11-s + 13-s − 5·17-s − 3·19-s − 9·23-s − 10·25-s − 16·29-s − 6·31-s + 2·37-s − 4·41-s + 8·43-s + 10·47-s + 3·49-s − 7·53-s + 6·59-s + 12·61-s − 18·67-s − 4·71-s − 13·73-s − 9·77-s − 2·79-s + 13·83-s − 89-s + 3·91-s − 8·97-s − 12·101-s + ⋯
L(s)  = 1  + 1.13·7-s − 0.904·11-s + 0.277·13-s − 1.21·17-s − 0.688·19-s − 1.87·23-s − 2·25-s − 2.97·29-s − 1.07·31-s + 0.328·37-s − 0.624·41-s + 1.21·43-s + 1.45·47-s + 3/7·49-s − 0.961·53-s + 0.781·59-s + 1.53·61-s − 2.19·67-s − 0.474·71-s − 1.52·73-s − 1.02·77-s − 0.225·79-s + 1.42·83-s − 0.105·89-s + 0.314·91-s − 0.812·97-s − 1.19·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7096896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7096896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7096896\)    =    \(2^{6} \cdot 3^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(452.504\)
Root analytic conductor: \(4.61217\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 7096896,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
37$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$D_{4}$ \( 1 - 3 T + 6 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.7.ad_g
11$C_2^2$ \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_o
13$D_{4}$ \( 1 - T + 16 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_q
17$D_{4}$ \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.17.f_be
19$D_{4}$ \( 1 + 3 T + 30 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.19.d_be
23$D_{4}$ \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.23.j_ce
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.29.q_es
31$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_be
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$D_{4}$ \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.47.ak_da
53$D_{4}$ \( 1 + 7 T + 108 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.53.h_ee
59$D_{4}$ \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_di
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$D_{4}$ \( 1 + 18 T + 174 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.67.s_gs
71$D_{4}$ \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.71.e_as
73$C_2^2$ \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.73.n_ds
79$D_{4}$ \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_eo
83$D_{4}$ \( 1 - 13 T + 198 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.83.an_hq
89$D_{4}$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) 2.89.b_ada
97$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_bu
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

−8.567598428071791194158469224599, −8.289783311521044366665344453758, −7.72188996798559198727845383307, −7.70191529641526886794879423072, −7.33431785892438045804946770728, −6.86939421316099375351207234994, −6.12284326729511163222092422687, −5.92606564445110629741047720468, −5.61402314452280825841055489009, −5.27107472717488115624239630137, −4.67138429429618325679406007318, −4.15235217996990448470009590320, −3.87499184353143999769580666374, −3.73994726428628278508621486636, −2.61771739227286233218441479705, −2.32367260952132751877427714794, −1.80507634182553979148765366520, −1.57100493842474426750906515857, 0, 0, 1.57100493842474426750906515857, 1.80507634182553979148765366520, 2.32367260952132751877427714794, 2.61771739227286233218441479705, 3.73994726428628278508621486636, 3.87499184353143999769580666374, 4.15235217996990448470009590320, 4.67138429429618325679406007318, 5.27107472717488115624239630137, 5.61402314452280825841055489009, 5.92606564445110629741047720468, 6.12284326729511163222092422687, 6.86939421316099375351207234994, 7.33431785892438045804946770728, 7.70191529641526886794879423072, 7.72188996798559198727845383307, 8.289783311521044366665344453758, 8.567598428071791194158469224599

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