Properties

 Label 4-338688-1.1-c1e2-0-8 Degree $4$ Conductor $338688$ Sign $1$ Analytic cond. $21.5950$ Root an. cond. $2.15570$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

Origins

Dirichlet series

 L(s)  = 1 − 3-s + 9-s − 4·13-s + 8·19-s − 2·25-s − 27-s + 4·37-s + 4·39-s + 49-s − 8·57-s − 4·61-s − 12·73-s + 2·75-s + 16·79-s + 81-s + 20·97-s − 4·109-s − 4·111-s − 4·117-s − 10·121-s + 127-s + 131-s + 137-s + 139-s − 147-s + 149-s + 151-s + ⋯
 L(s)  = 1 − 0.577·3-s + 1/3·9-s − 1.10·13-s + 1.83·19-s − 2/5·25-s − 0.192·27-s + 0.657·37-s + 0.640·39-s + 1/7·49-s − 1.05·57-s − 0.512·61-s − 1.40·73-s + 0.230·75-s + 1.80·79-s + 1/9·81-s + 2.03·97-s − 0.383·109-s − 0.379·111-s − 0.369·117-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0824·147-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$4$$ Conductor: $$338688$$    =    $$2^{8} \cdot 3^{3} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$21.5950$$ Root analytic conductor: $$2.15570$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 338688,\ (\ :1/2, 1/2),\ 1)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$1.291768351$$ $$L(\frac12)$$ $$\approx$$ $$1.291768351$$ $$L(\frac{3}{2})$$ 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$C_1$ $$1 + T$$
7$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good5$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
11$C_2^2$ $$1 + 10 T^{2} + p^{2} T^{4}$$
13$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
17$C_2^2$ $$1 + 26 T^{2} + p^{2} T^{4}$$
19$C_2$$\times$$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + p T^{2} )$$
23$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
29$C_2^2$ $$1 + 38 T^{2} + p^{2} T^{4}$$
31$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
37$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
41$C_2^2$ $$1 - 22 T^{2} + p^{2} T^{4}$$
43$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
47$C_2^2$ $$1 - 18 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 - 58 T^{2} + p^{2} T^{4}$$
59$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
67$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
71$C_2^2$ $$1 + 66 T^{2} + p^{2} T^{4}$$
73$C_2$$\times$$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
79$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
83$C_2^2$ $$1 - 106 T^{2} + p^{2} T^{4}$$
89$C_2^2$ $$1 + 106 T^{2} + p^{2} T^{4}$$
97$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
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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

−8.925045457299704603116489065928, −8.079277740952760928945829672096, −7.79767231908533927131642959225, −7.32162986839021161513702366232, −7.00255343044110607173025669696, −6.37247888227728798454797289956, −5.81505070643734415933143757401, −5.46167295341240608742547978354, −4.84778883354414629110762089589, −4.55847983479225625423580151782, −3.74996468035003867898347641963, −3.16913179166411359156679433175, −2.51551692446386328944873714774, −1.68169268545289806799009243015, −0.68580780236667830437631697436, 0.68580780236667830437631697436, 1.68169268545289806799009243015, 2.51551692446386328944873714774, 3.16913179166411359156679433175, 3.74996468035003867898347641963, 4.55847983479225625423580151782, 4.84778883354414629110762089589, 5.46167295341240608742547978354, 5.81505070643734415933143757401, 6.37247888227728798454797289956, 7.00255343044110607173025669696, 7.32162986839021161513702366232, 7.79767231908533927131642959225, 8.079277740952760928945829672096, 8.925045457299704603116489065928