Properties

Label 4-114e2-1.1-c1e2-0-13
Degree $4$
Conductor $12996$
Sign $1$
Analytic cond. $0.828636$
Root an. cond. $0.954093$
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 − 3·3-s − 6·5-s + 3·6-s − 4·7-s + 8-s + 6·9-s + 6·10-s − 6·13-s + 4·14-s + 18·15-s − 16-s − 12·17-s − 6·18-s + 19-s + 12·21-s − 3·24-s + 19·25-s + 6·26-s − 9·27-s + 6·29-s − 18·30-s + 12·34-s + 24·35-s − 38-s + 18·39-s − 6·40-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s − 2.68·5-s + 1.22·6-s − 1.51·7-s + 0.353·8-s + 2·9-s + 1.89·10-s − 1.66·13-s + 1.06·14-s + 4.64·15-s − 1/4·16-s − 2.91·17-s − 1.41·18-s + 0.229·19-s + 2.61·21-s − 0.612·24-s + 19/5·25-s + 1.17·26-s − 1.73·27-s + 1.11·29-s − 3.28·30-s + 2.05·34-s + 4.05·35-s − 0.162·38-s + 2.88·39-s − 0.948·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12996\)    =    \(2^{2} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.828636\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{114} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 12996,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 + p T + p T^{2} \)
19$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 9 T + 94 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 5 T - 48 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 24 T + 271 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 139 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
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

−12.90827632006148094617876878745, −12.64180788202229429145452644977, −12.10867606907445232844022519677, −11.76088000692072397196604889310, −11.18412967965461406980137132853, −11.01410673724088045363400974489, −10.20701896490707379654839101008, −9.817584436886253214074820769885, −8.998245159580685822868552706014, −8.522355992659609080579137339979, −7.56110063126240219339484692946, −7.34965980248173268667283238865, −6.57667224547453187125174561201, −6.46395096170721997174046984607, −4.99762765579652302841456530831, −4.50093324347433417577487718006, −4.09377601857051945711254367337, −2.92474174787155679446208324715, 0, 0, 2.92474174787155679446208324715, 4.09377601857051945711254367337, 4.50093324347433417577487718006, 4.99762765579652302841456530831, 6.46395096170721997174046984607, 6.57667224547453187125174561201, 7.34965980248173268667283238865, 7.56110063126240219339484692946, 8.522355992659609080579137339979, 8.998245159580685822868552706014, 9.817584436886253214074820769885, 10.20701896490707379654839101008, 11.01410673724088045363400974489, 11.18412967965461406980137132853, 11.76088000692072397196604889310, 12.10867606907445232844022519677, 12.64180788202229429145452644977, 12.90827632006148094617876878745

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