Properties

Label 4-40e4-1.1-c1e2-0-15
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 + 2·9-s − 2·11-s + 2·13-s + 4·17-s − 6·19-s − 6·27-s + 6·29-s + 16·31-s + 4·33-s − 6·37-s − 4·39-s + 10·43-s + 16·47-s + 10·49-s − 8·51-s + 10·53-s + 12·57-s + 6·59-s − 18·61-s − 10·67-s + 11·81-s − 2·83-s − 12·87-s − 32·93-s + 4·97-s − 4·99-s + ⋯
L(s)  = 1  − 1.15·3-s + 2/3·9-s − 0.603·11-s + 0.554·13-s + 0.970·17-s − 1.37·19-s − 1.15·27-s + 1.11·29-s + 2.87·31-s + 0.696·33-s − 0.986·37-s − 0.640·39-s + 1.52·43-s + 2.33·47-s + 10/7·49-s − 1.12·51-s + 1.37·53-s + 1.58·57-s + 0.781·59-s − 2.30·61-s − 1.22·67-s + 11/9·81-s − 0.219·83-s − 1.28·87-s − 3.31·93-s + 0.406·97-s − 0.402·99-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: induced by $\chi_{1600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.480508866\)
\(L(\frac12)\) \(\approx\) \(1.480508866\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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

−9.872932570322977685621864201960, −9.055087284914310541743538158781, −8.741725797543948219256329406089, −8.599073138777116856074738628014, −7.895540715125077742601543027210, −7.56250096929534773600998650581, −7.28393738337688952010220885128, −6.63568898522911555025553376384, −6.20228723236467261597295882552, −6.03780032195121777799909554856, −5.56488020806183165662227679280, −5.23603551650315258810324566151, −4.47398920521395097767225012078, −4.36555707860134889009954967928, −3.85977087013884148245362238782, −3.05563059111024567756532877868, −2.61337118097625187623038090185, −2.03026736438775170760413857794, −1.05445676113746192266653822891, −0.62372294225937580169693360239, 0.62372294225937580169693360239, 1.05445676113746192266653822891, 2.03026736438775170760413857794, 2.61337118097625187623038090185, 3.05563059111024567756532877868, 3.85977087013884148245362238782, 4.36555707860134889009954967928, 4.47398920521395097767225012078, 5.23603551650315258810324566151, 5.56488020806183165662227679280, 6.03780032195121777799909554856, 6.20228723236467261597295882552, 6.63568898522911555025553376384, 7.28393738337688952010220885128, 7.56250096929534773600998650581, 7.895540715125077742601543027210, 8.599073138777116856074738628014, 8.741725797543948219256329406089, 9.055087284914310541743538158781, 9.872932570322977685621864201960

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