Properties

 Label 4-117e2-1.1-c1e2-0-9 Degree $4$ Conductor $13689$ Sign $1$ Analytic cond. $0.872822$ Root an. cond. $0.966565$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

Origins

Dirichlet series

 L(s)  = 1 + 3-s + 2·4-s − 2·9-s + 2·12-s + 5·13-s − 6·17-s − 3·23-s − 25-s − 5·27-s + 6·29-s − 4·36-s + 5·39-s − 11·43-s + 5·49-s − 6·51-s + 10·52-s − 6·53-s + 13·61-s − 8·64-s − 12·68-s − 3·69-s − 75-s + 7·79-s + 81-s + 6·87-s − 6·92-s − 2·100-s + ⋯
 L(s)  = 1 + 0.577·3-s + 4-s − 2/3·9-s + 0.577·12-s + 1.38·13-s − 1.45·17-s − 0.625·23-s − 1/5·25-s − 0.962·27-s + 1.11·29-s − 2/3·36-s + 0.800·39-s − 1.67·43-s + 5/7·49-s − 0.840·51-s + 1.38·52-s − 0.824·53-s + 1.66·61-s − 64-s − 1.45·68-s − 0.361·69-s − 0.115·75-s + 0.787·79-s + 1/9·81-s + 0.643·87-s − 0.625·92-s − 1/5·100-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$4$$ Conductor: $$13689$$    =    $$3^{4} \cdot 13^{2}$$ Sign: $1$ Analytic conductor: $$0.872822$$ Root analytic conductor: $$0.966565$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 13689,\ (\ :1/2, 1/2),\ 1)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$1.462791507$$ $$L(\frac12)$$ $$\approx$$ $$1.462791507$$ $$L(\frac{3}{2})$$ 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)$
bad3$C_2$ $$1 - T + p T^{2}$$
13$C_2$ $$1 - 5 T + p T^{2}$$
good2$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
5$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
7$C_2^2$ $$1 - 5 T^{2} + p^{2} T^{4}$$
11$C_2^2$ $$1 - 5 T^{2} + p^{2} T^{4}$$
17$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
19$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
23$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 3 T + p T^{2} )$$
29$C_2$$\times$$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
31$C_2^2$ $$1 + 55 T^{2} + p^{2} T^{4}$$
37$C_2^2$ $$1 + 34 T^{2} + p^{2} T^{4}$$
41$C_2^2$ $$1 + 37 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
47$C_2^2$ $$1 + 55 T^{2} + p^{2} T^{4}$$
53$C_2$$\times$$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
59$C_2^2$ $$1 - 53 T^{2} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
67$C_2^2$ $$1 - 89 T^{2} + p^{2} T^{4}$$
71$C_2^2$ $$1 - 98 T^{2} + p^{2} T^{4}$$
73$C_2^2$ $$1 + 34 T^{2} + p^{2} T^{4}$$
79$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} )$$
83$C_2^2$ $$1 - 125 T^{2} + p^{2} T^{4}$$
89$C_2^2$ $$1 + 130 T^{2} + p^{2} T^{4}$$
97$C_2^2$ $$1 + 157 T^{2} + p^{2} T^{4}$$
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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

−11.06937929645315365520178910784, −10.93594436375944591737145655664, −10.22169940461072532546528114430, −9.497585477585207081976371047729, −8.871279211368385913124115833839, −8.373990587725152631748713218362, −8.059360947468859413416339897163, −7.10090601050224016178138442551, −6.53789513740480786554803192237, −6.19102450579877077584468932372, −5.34699002199756816358806463821, −4.34760802050208508977310815179, −3.54424435234633271901561057637, −2.69293404486165492040435902402, −1.89344939171266350594104110479, 1.89344939171266350594104110479, 2.69293404486165492040435902402, 3.54424435234633271901561057637, 4.34760802050208508977310815179, 5.34699002199756816358806463821, 6.19102450579877077584468932372, 6.53789513740480786554803192237, 7.10090601050224016178138442551, 8.059360947468859413416339897163, 8.373990587725152631748713218362, 8.871279211368385913124115833839, 9.497585477585207081976371047729, 10.22169940461072532546528114430, 10.93594436375944591737145655664, 11.06937929645315365520178910784