# Properties

 Degree $4$ Conductor $2480625$ Sign $1$ Motivic weight $0$ Primitive no Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 2·4-s + 3·16-s − 49-s + 4·64-s − 4·79-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + 197-s + 199-s + 211-s + 223-s + ⋯
 L(s)  = 1 + 2·4-s + 3·16-s − 49-s + 4·64-s − 4·79-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 2·196-s + 197-s + 199-s + 211-s + 223-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$2480625$$    =    $$3^{4} \cdot 5^{4} \cdot 7^{2}$$ Sign: $1$ Motivic weight: $$0$$ Character: induced by $\chi_{1575} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 2480625,\ (\ :0, 0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.976546573$$ $$L(\frac12)$$ $$\approx$$ $$1.976546573$$ $$L(1)$$ 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 $$1$$
5 $$1$$
7$C_2$ $$1 + T^{2}$$
good2$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
11$C_2$ $$( 1 + T^{2} )^{2}$$
13$C_2$ $$( 1 + T^{2} )^{2}$$
17$C_2$ $$( 1 + T^{2} )^{2}$$
19$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
23$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
29$C_2$ $$( 1 + T^{2} )^{2}$$
31$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
37$C_2$ $$( 1 + T^{2} )^{2}$$
41$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
43$C_2$ $$( 1 + T^{2} )^{2}$$
47$C_2$ $$( 1 + T^{2} )^{2}$$
53$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
59$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
61$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
67$C_2$ $$( 1 + T^{2} )^{2}$$
71$C_2$ $$( 1 + T^{2} )^{2}$$
73$C_2$ $$( 1 + T^{2} )^{2}$$
79$C_1$ $$( 1 + T )^{4}$$
83$C_2$ $$( 1 + T^{2} )^{2}$$
89$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
97$C_2$ $$( 1 + T^{2} )^{2}$$
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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

−9.777662703165187287566671656296, −9.751939668581427896467730196113, −8.785094445161926091883884528579, −8.696821163695744809947302107927, −8.120982520058901001726358376490, −7.69533806277893102285417128286, −7.25543411897011803892344680586, −7.20254116341639988099969785528, −6.42567958369940700772505248296, −6.41633951449973542666162314434, −5.72494990932537743023290213341, −5.62711434928389653058091435628, −4.85669050903742721122450969610, −4.44484030762349950054090385695, −3.53510049815431303241984854106, −3.47715774174445045269970651000, −2.61732237927340724392963698213, −2.49888726773389951467309249421, −1.66337124634595529707566199574, −1.27033692199437885483573765158, 1.27033692199437885483573765158, 1.66337124634595529707566199574, 2.49888726773389951467309249421, 2.61732237927340724392963698213, 3.47715774174445045269970651000, 3.53510049815431303241984854106, 4.44484030762349950054090385695, 4.85669050903742721122450969610, 5.62711434928389653058091435628, 5.72494990932537743023290213341, 6.41633951449973542666162314434, 6.42567958369940700772505248296, 7.20254116341639988099969785528, 7.25543411897011803892344680586, 7.69533806277893102285417128286, 8.120982520058901001726358376490, 8.696821163695744809947302107927, 8.785094445161926091883884528579, 9.751939668581427896467730196113, 9.777662703165187287566671656296