# Properties

 Label 4-245e2-1.1-c1e2-0-9 Degree $4$ Conductor $60025$ Sign $-1$ Analytic cond. $3.82724$ Root an. cond. $1.39869$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $1$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·2-s + 8·4-s − 8·8-s + 3·9-s + 2·11-s − 4·16-s − 12·18-s − 8·22-s − 8·23-s + 25-s − 2·29-s + 32·32-s + 24·36-s − 12·43-s + 16·44-s + 32·46-s − 4·50-s − 20·53-s + 8·58-s − 64·64-s − 28·67-s − 16·71-s − 24·72-s − 2·79-s + 48·86-s − 16·88-s − 64·92-s + ⋯
 L(s)  = 1 − 2.82·2-s + 4·4-s − 2.82·8-s + 9-s + 0.603·11-s − 16-s − 2.82·18-s − 1.70·22-s − 1.66·23-s + 1/5·25-s − 0.371·29-s + 5.65·32-s + 4·36-s − 1.82·43-s + 2.41·44-s + 4.71·46-s − 0.565·50-s − 2.74·53-s + 1.05·58-s − 8·64-s − 3.42·67-s − 1.89·71-s − 2.82·72-s − 0.225·79-s + 5.17·86-s − 1.70·88-s − 6.67·92-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$60025$$    =    $$5^{2} \cdot 7^{4}$$ Sign: $-1$ Analytic conductor: $$3.82724$$ Root analytic conductor: $$1.39869$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 60025,\ (\ :1/2, 1/2),\ -1)$$

## 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)$
bad5$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
7 $$1$$
good2$C_2$ $$( 1 + p T + p T^{2} )^{2}$$
3$C_2$ $$( 1 - p T + p T^{2} )( 1 + p T + p T^{2} )$$
11$C_2$ $$( 1 - T + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
17$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
19$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
23$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
29$C_2$ $$( 1 + T + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
37$C_2$ $$( 1 + p T^{2} )^{2}$$
41$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
43$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
47$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
53$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
59$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
61$C_2$ $$( 1 + p T^{2} )^{2}$$
67$C_2$ $$( 1 + 14 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
79$C_2$ $$( 1 + T + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
89$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
97$C_2$ $$( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} )$$
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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

−9.674071857485304380380573397765, −9.154994581895650846359851030198, −8.895041085769743028052283031023, −8.210184381994137443519230420468, −7.79509008882088804636414623325, −7.56848203557801886905576002740, −6.67134380816033742861497166759, −6.66812515626338690414111791742, −5.69074574951383439724683067976, −4.49258853804675338518966926948, −4.33023383010768085745155570778, −3.07565995342284756040604408072, −1.70898778880961935771596537039, −1.56775892122303114558742737725, 0, 1.56775892122303114558742737725, 1.70898778880961935771596537039, 3.07565995342284756040604408072, 4.33023383010768085745155570778, 4.49258853804675338518966926948, 5.69074574951383439724683067976, 6.66812515626338690414111791742, 6.67134380816033742861497166759, 7.56848203557801886905576002740, 7.79509008882088804636414623325, 8.210184381994137443519230420468, 8.895041085769743028052283031023, 9.154994581895650846359851030198, 9.674071857485304380380573397765