# Properties

 Label 4-930e2-1.1-c1e2-0-17 Degree $4$ Conductor $864900$ Sign $-1$ Analytic cond. $55.1467$ Root an. cond. $2.72508$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Learn more

## Dirichlet series

 L(s)  = 1 − 2·3-s − 4-s + 6·7-s + 9-s + 2·12-s + 2·13-s + 16-s − 12·21-s + 25-s + 4·27-s − 6·28-s − 36-s − 10·37-s − 4·39-s − 12·43-s − 2·48-s + 14·49-s − 2·52-s − 4·61-s + 6·63-s − 64-s − 6·67-s + 22·73-s − 2·75-s − 11·81-s + 12·84-s + 12·91-s + ⋯
 L(s)  = 1 − 1.15·3-s − 1/2·4-s + 2.26·7-s + 1/3·9-s + 0.577·12-s + 0.554·13-s + 1/4·16-s − 2.61·21-s + 1/5·25-s + 0.769·27-s − 1.13·28-s − 1/6·36-s − 1.64·37-s − 0.640·39-s − 1.82·43-s − 0.288·48-s + 2·49-s − 0.277·52-s − 0.512·61-s + 0.755·63-s − 1/8·64-s − 0.733·67-s + 2.57·73-s − 0.230·75-s − 1.22·81-s + 1.30·84-s + 1.25·91-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$864900$$    =    $$2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}$$ Sign: $-1$ Analytic conductor: $$55.1467$$ Root analytic conductor: $$2.72508$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 864900,\ (\ :1/2, 1/2),\ -1)$$

## 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$C_2$ $$1 + T^{2}$$
3$C_2$ $$1 + 2 T + p T^{2}$$
5$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
31$C_2$ $$1 + p T^{2}$$
good7$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
11$C_2^2$ $$1 + 10 T^{2} + p^{2} T^{4}$$
13$C_2$$\times$$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
19$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
23$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
29$C_2^2$ $$1 - 22 T^{2} + p^{2} T^{4}$$
37$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 10 T + p T^{2} )$$
41$C_2^2$ $$1 + 42 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 12 T + p T^{2} )$$
47$C_2^2$ $$1 + 10 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
59$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
61$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
67$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 6 T + p T^{2} )$$
71$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
73$C_2$$\times$$C_2$ $$( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} )$$
79$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
83$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
89$C_2^2$ $$1 - 102 T^{2} + p^{2} T^{4}$$
97$C_2$$\times$$C_2$ $$( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
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

−8.094516989884481716872748546888, −7.70158203757975610770315706406, −6.93218152688057401872567142051, −6.68770055729895198216161364580, −6.14970985396899297248183326190, −5.37148186282783477747388186443, −5.29117441073688713850944236676, −4.97630027277575908068295367977, −4.44514379194632604451878223603, −3.94426177116844605090509920301, −3.30908920407833803053253029557, −2.42989353523414900304035353582, −1.53347833820744832754504858839, −1.28435873278236245967725570035, 0, 1.28435873278236245967725570035, 1.53347833820744832754504858839, 2.42989353523414900304035353582, 3.30908920407833803053253029557, 3.94426177116844605090509920301, 4.44514379194632604451878223603, 4.97630027277575908068295367977, 5.29117441073688713850944236676, 5.37148186282783477747388186443, 6.14970985396899297248183326190, 6.68770055729895198216161364580, 6.93218152688057401872567142051, 7.70158203757975610770315706406, 8.094516989884481716872748546888