Properties

Label 4-457056-1.1-c1e2-0-19
Degree $4$
Conductor $457056$
Sign $1$
Analytic cond. $29.1422$
Root an. cond. $2.32343$
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  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 12·11-s − 12-s − 4·13-s + 16-s − 18-s + 12·22-s − 2·23-s + 24-s − 6·25-s + 4·26-s − 27-s − 32-s + 12·33-s + 36-s + 4·39-s − 12·44-s + 2·46-s − 16·47-s − 48-s − 10·49-s + 6·50-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 3.61·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.235·18-s + 2.55·22-s − 0.417·23-s + 0.204·24-s − 6/5·25-s + 0.784·26-s − 0.192·27-s − 0.176·32-s + 2.08·33-s + 1/6·36-s + 0.640·39-s − 1.80·44-s + 0.294·46-s − 2.33·47-s − 0.144·48-s − 1.42·49-s + 0.848·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 457056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 457056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(457056\)    =    \(2^{5} \cdot 3^{3} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(29.1422\)
Root analytic conductor: \(2.32343\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 457056,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( 1 + T \)
3$C_1$ \( 1 + T \)
23$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−8.009088544194373664704576673472, −7.74940445665935159391608281760, −7.48954786032446015166944686975, −6.81404055785242482314012546704, −6.25250569926844252670184221360, −5.70446099983308366623422114205, −5.34329120767643148260995938996, −4.78570582427944795062347753374, −4.61781233004508478544485198077, −3.38807414963777119141033594454, −2.91631229938310047782795220333, −2.33137889789155181282145246786, −1.75673035985210661335841948053, 0, 0, 1.75673035985210661335841948053, 2.33137889789155181282145246786, 2.91631229938310047782795220333, 3.38807414963777119141033594454, 4.61781233004508478544485198077, 4.78570582427944795062347753374, 5.34329120767643148260995938996, 5.70446099983308366623422114205, 6.25250569926844252670184221360, 6.81404055785242482314012546704, 7.48954786032446015166944686975, 7.74940445665935159391608281760, 8.009088544194373664704576673472

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