Properties

Label 4-2592e2-1.1-c1e2-0-38
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 6·13-s − 8·17-s − 7·25-s − 2·29-s − 16·37-s − 10·41-s − 11·49-s + 16·53-s − 14·61-s − 12·65-s − 24·73-s − 16·85-s + 8·89-s − 6·97-s + 26·101-s + 2·113-s − 19·121-s − 26·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.66·13-s − 1.94·17-s − 7/5·25-s − 0.371·29-s − 2.63·37-s − 1.56·41-s − 1.57·49-s + 2.19·53-s − 1.79·61-s − 1.48·65-s − 2.80·73-s − 1.73·85-s + 0.847·89-s − 0.609·97-s + 2.58·101-s + 0.188·113-s − 1.72·121-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.332·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-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)$
bad2 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 115 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 59 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 131 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 91 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.637345944703892956778720718646, −8.601405563217285686555412492056, −7.75128918025107996069174544630, −7.60458059561689485714044977973, −6.97541285295661652516503663036, −6.96846308061153964022842135026, −6.27281084300620682416424017005, −6.14836806197630852271087800199, −5.44948790055200400199659619793, −5.21014053283776002087142178793, −4.81213860529183161328166341937, −4.40438572519139842235840976944, −3.88341084047931408724670128105, −3.38033479377950849268674960368, −2.78684122154950677537604244863, −2.27571844436291562708477831570, −1.86485531322756247966908211949, −1.58663246186762614829046343021, 0, 0, 1.58663246186762614829046343021, 1.86485531322756247966908211949, 2.27571844436291562708477831570, 2.78684122154950677537604244863, 3.38033479377950849268674960368, 3.88341084047931408724670128105, 4.40438572519139842235840976944, 4.81213860529183161328166341937, 5.21014053283776002087142178793, 5.44948790055200400199659619793, 6.14836806197630852271087800199, 6.27281084300620682416424017005, 6.96846308061153964022842135026, 6.97541285295661652516503663036, 7.60458059561689485714044977973, 7.75128918025107996069174544630, 8.601405563217285686555412492056, 8.637345944703892956778720718646

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