Properties

Label 4-3456e2-1.1-c1e2-0-7
Degree $4$
Conductor $11943936$
Sign $1$
Analytic cond. $761.555$
Root an. cond. $5.25321$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 2·7-s − 5·11-s + 4·13-s − 2·17-s + 10·19-s + 4·23-s + 5·25-s + 6·29-s − 4·35-s + 20·37-s − 3·41-s − 9·43-s − 8·47-s + 7·49-s − 24·53-s − 10·55-s − 7·59-s + 4·61-s + 8·65-s − 7·67-s + 12·71-s − 26·73-s + 10·77-s + 2·79-s − 12·83-s − 4·85-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.755·7-s − 1.50·11-s + 1.10·13-s − 0.485·17-s + 2.29·19-s + 0.834·23-s + 25-s + 1.11·29-s − 0.676·35-s + 3.28·37-s − 0.468·41-s − 1.37·43-s − 1.16·47-s + 49-s − 3.29·53-s − 1.34·55-s − 0.911·59-s + 0.512·61-s + 0.992·65-s − 0.855·67-s + 1.42·71-s − 3.04·73-s + 1.13·77-s + 0.225·79-s − 1.31·83-s − 0.433·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11943936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11943936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11943936\)    =    \(2^{14} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(761.555\)
Root analytic conductor: \(5.25321\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11943936,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.610241737\)
\(L(\frac12)\) \(\approx\) \(2.610241737\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 2 T - 75 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
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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

−9.078222163131816112069864243488, −8.468983929463685864292328473609, −7.979694558531705259051552474540, −7.64269541855195498322632196965, −7.41290901549363365020360540308, −6.80792810469153659363105641886, −6.48394142713119506769567954406, −6.12764165941645074112470275210, −5.86569305112497598767228733479, −5.37866413469574304479809989120, −5.05169353119844772717967545268, −4.57372404172688651808715723050, −4.36959719898872370933143702679, −3.37341408085837235225725460641, −3.11723256993779415476941415577, −2.90291433819347596935626055566, −2.52002622521125681170746608067, −1.54569403600727640527068591547, −1.29853942712331482999645281952, −0.50956696796519522620260550200, 0.50956696796519522620260550200, 1.29853942712331482999645281952, 1.54569403600727640527068591547, 2.52002622521125681170746608067, 2.90291433819347596935626055566, 3.11723256993779415476941415577, 3.37341408085837235225725460641, 4.36959719898872370933143702679, 4.57372404172688651808715723050, 5.05169353119844772717967545268, 5.37866413469574304479809989120, 5.86569305112497598767228733479, 6.12764165941645074112470275210, 6.48394142713119506769567954406, 6.80792810469153659363105641886, 7.41290901549363365020360540308, 7.64269541855195498322632196965, 7.979694558531705259051552474540, 8.468983929463685864292328473609, 9.078222163131816112069864243488

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