Properties

Label 4-3456e2-1.1-c1e2-0-23
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 $2$

Origins

Origins of factors

Downloads

Learn more

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: \(2\)
Selberg data: \((4,\ 11943936,\ (\ :1/2, 1/2),\ 1)\)

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$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
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

−8.452576685040873648601349959631, −8.257867809038482960450034760515, −7.42097593678275114819297366576, −7.27288845220825905730133330062, −6.82607837403655934154670228704, −6.63873940217139958063592763453, −6.50833163270648095725746696478, −5.68564364598820840199351247046, −5.29335547886563723879482722787, −5.10263849895104316158316848147, −4.20029605006723500991586844923, −4.16938209427224475275731329436, −3.83654964063556896742550011938, −3.43980589976707906914659552306, −2.58457084481365404086027358371, −2.51522742879910151141871075036, −1.73478539784954574970057285916, −1.16022226607341470172710149870, 0, 0, 1.16022226607341470172710149870, 1.73478539784954574970057285916, 2.51522742879910151141871075036, 2.58457084481365404086027358371, 3.43980589976707906914659552306, 3.83654964063556896742550011938, 4.16938209427224475275731329436, 4.20029605006723500991586844923, 5.10263849895104316158316848147, 5.29335547886563723879482722787, 5.68564364598820840199351247046, 6.50833163270648095725746696478, 6.63873940217139958063592763453, 6.82607837403655934154670228704, 7.27288845220825905730133330062, 7.42097593678275114819297366576, 8.257867809038482960450034760515, 8.452576685040873648601349959631

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