↑

Properties

Label 4-2016e2-1.1-c1e2-0-14
Degree $4$
Conductor $4064256$
Sign $1$
Analytic cond. $259.140$
Root an. cond. $4.01221$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 2·7-s − 2·11-s + 6·17-s + 4·19-s − 2·23-s − 2·25-s + 12·29-s − 4·31-s − 4·35-s + 10·41-s + 8·43-s + 12·47-s + 3·49-s + 16·53-s − 4·55-s − 4·59-s + 16·67-s + 10·71-s − 12·73-s + 4·77-s − 8·79-s + 8·83-s + 12·85-s + 18·89-s + 8·95-s + 4·97-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.755·7-s − 0.603·11-s + 1.45·17-s + 0.917·19-s − 0.417·23-s − 2/5·25-s + 2.22·29-s − 0.718·31-s − 0.676·35-s + 1.56·41-s + 1.21·43-s + 1.75·47-s + 3/7·49-s + 2.19·53-s − 0.539·55-s − 0.520·59-s + 1.95·67-s + 1.18·71-s − 1.40·73-s + 0.455·77-s − 0.900·79-s + 0.878·83-s + 1.30·85-s + 1.90·89-s + 0.820·95-s + 0.406·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4064256\)    =    \(2^{10} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(259.140\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4064256,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.206017685\)
\(L(\frac12)\) \(\approx\) \(3.206017685\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_g
11$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_s
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_bm
19$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_w
23$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_c
29$C_4$ \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.29.am_cw
31$C_4$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_bu
37$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.37.a_cc
41$D_{4}$ \( 1 - 10 T + 102 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.41.ak_dy
43$D_{4}$ \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.43.ai_w
47$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_eg
53$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.53.aq_go
59$D_{4}$ \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_dy
61$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \) 2.61.a_dy
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$D_{4}$ \( 1 - 10 T + 162 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.71.ak_gg
73$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.73.m_dy
79$D_{4}$ \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_dq
83$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.83.ai_dy
89$D_{4}$ \( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.89.as_ig
97$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_eo
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

−9.332976879480965946339801318570, −9.134834858892614840915914679651, −8.594643059582245032868058279050, −8.163617449054383343278519250470, −7.63992853172979644479414117555, −7.55248573413277655306455285828, −6.95840846181893149317189402437, −6.61613684007233408689804192440, −5.91047112268785372026047282921, −5.89882568263470171560901893075, −5.39632904969333878206191003883, −5.18940926167781032265455503611, −4.29553804527633880514140591163, −4.11838341146507871112741868173, −3.36959227546149659898609422955, −3.05970431948967431263468952973, −2.31343640255533579327817331562, −2.28092617025967857993274484497, −1.06235196571297911668335168850, −0.78484562974039876332005764318, 0.78484562974039876332005764318, 1.06235196571297911668335168850, 2.28092617025967857993274484497, 2.31343640255533579327817331562, 3.05970431948967431263468952973, 3.36959227546149659898609422955, 4.11838341146507871112741868173, 4.29553804527633880514140591163, 5.18940926167781032265455503611, 5.39632904969333878206191003883, 5.89882568263470171560901893075, 5.91047112268785372026047282921, 6.61613684007233408689804192440, 6.95840846181893149317189402437, 7.55248573413277655306455285828, 7.63992853172979644479414117555, 8.163617449054383343278519250470, 8.594643059582245032868058279050, 9.134834858892614840915914679651, 9.332976879480965946339801318570

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