↑

Properties

Label 4-40e4-1.1-c1e2-0-14
Degree $4$
Conductor $2560000$
Sign $1$
Analytic cond. $163.227$
Root an. cond. $3.57436$
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  − 2·3-s + 2·7-s + 2·9-s + 8·13-s + 8·17-s − 8·19-s − 4·21-s + 10·23-s − 6·27-s − 16·39-s − 8·41-s − 14·43-s + 6·47-s + 2·49-s − 16·51-s + 8·53-s + 16·57-s − 8·59-s + 16·61-s + 4·63-s + 6·67-s − 20·69-s − 8·73-s − 16·79-s + 11·81-s + 10·83-s + 16·91-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.755·7-s + 2/3·9-s + 2.21·13-s + 1.94·17-s − 1.83·19-s − 0.872·21-s + 2.08·23-s − 1.15·27-s − 2.56·39-s − 1.24·41-s − 2.13·43-s + 0.875·47-s + 2/7·49-s − 2.24·51-s + 1.09·53-s + 2.11·57-s − 1.04·59-s + 2.04·61-s + 0.503·63-s + 0.733·67-s − 2.40·69-s − 0.936·73-s − 1.80·79-s + 11/9·81-s + 1.09·83-s + 1.67·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2560000\)    =    \(2^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(163.227\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.835157122\)
\(L(\frac12)\) \(\approx\) \(1.835157122\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_c
7$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_c
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.13.ai_bg
17$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.17.ai_bg
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.23.ak_by
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.29.a_acc
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2^2$ \( 1 + p^{2} T^{4} \) 2.37.a_a
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.41.i_du
43$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.43.o_du
47$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_s
53$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_bg
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.61.aq_he
67$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_s
71$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \) 2.71.a_ek
73$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_bg
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.83.ak_by
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.a_ada
97$C_2^2$ \( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.97.y_lc
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

−9.799213305508140722053793563820, −8.933674618950114818973548524953, −8.892890613355382179745385989393, −8.288093545460029554950883936100, −8.244102082229293383318882509651, −7.58955111222279669233963023116, −7.11515040985469529459204446923, −6.57925370113038588392895013871, −6.49187949885099807548182026998, −5.85807656234190864984834218509, −5.37907733006028821695365489587, −5.35600697270186469901385472790, −4.76900576626099330319206888718, −4.04886027243127850007050633577, −3.82921008566749396781685750606, −3.28228076936687235527126775812, −2.63970096018183434443584677962, −1.52306688034955979937522293970, −1.48924525087183715637078100899, −0.62363932292561094271429362883, 0.62363932292561094271429362883, 1.48924525087183715637078100899, 1.52306688034955979937522293970, 2.63970096018183434443584677962, 3.28228076936687235527126775812, 3.82921008566749396781685750606, 4.04886027243127850007050633577, 4.76900576626099330319206888718, 5.35600697270186469901385472790, 5.37907733006028821695365489587, 5.85807656234190864984834218509, 6.49187949885099807548182026998, 6.57925370113038588392895013871, 7.11515040985469529459204446923, 7.58955111222279669233963023116, 8.244102082229293383318882509651, 8.288093545460029554950883936100, 8.892890613355382179745385989393, 8.933674618950114818973548524953, 9.799213305508140722053793563820

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