# Properties

 Label 2-7200-1.1-c1-0-18 Degree $2$ Conductor $7200$ Sign $1$ Analytic cond. $57.4922$ Root an. cond. $7.58236$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 4·13-s − 8·17-s − 10·29-s + 12·37-s + 10·41-s − 7·49-s + 4·53-s + 10·61-s + 16·73-s + 10·89-s − 8·97-s + 2·101-s + 6·109-s + 16·113-s + ⋯
 L(s)  = 1 + 1.10·13-s − 1.94·17-s − 1.85·29-s + 1.97·37-s + 1.56·41-s − 49-s + 0.549·53-s + 1.28·61-s + 1.87·73-s + 1.05·89-s − 0.812·97-s + 0.199·101-s + 0.574·109-s + 1.50·113-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$7200$$    =    $$2^{5} \cdot 3^{2} \cdot 5^{2}$$ Sign: $1$ Analytic conductor: $$57.4922$$ Root analytic conductor: $$7.58236$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{7200} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 7200,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.810981779$$ $$L(\frac12)$$ $$\approx$$ $$1.810981779$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1$$
good7 $$1 + p T^{2}$$
11 $$1 + p T^{2}$$
13 $$1 - 4 T + p T^{2}$$
17 $$1 + 8 T + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 + 10 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 - 12 T + p T^{2}$$
41 $$1 - 10 T + p T^{2}$$
43 $$1 + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 - 4 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 - 10 T + p T^{2}$$
67 $$1 + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 - 16 T + p T^{2}$$
79 $$1 + p T^{2}$$
83 $$1 + p T^{2}$$
89 $$1 - 10 T + p T^{2}$$
97 $$1 + 8 T + p T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−7.935384661251128756487087682079, −7.23992932349289750072857579645, −6.40341818141713182323725618147, −6.00178162513688601397902655350, −5.08483969514461699201315648855, −4.20653677488776657079886207199, −3.74502694198768108207542352534, −2.60329712062717383983860995277, −1.88203915631401830676702420061, −0.67503075954131059935836478299, 0.67503075954131059935836478299, 1.88203915631401830676702420061, 2.60329712062717383983860995277, 3.74502694198768108207542352534, 4.20653677488776657079886207199, 5.08483969514461699201315648855, 6.00178162513688601397902655350, 6.40341818141713182323725618147, 7.23992932349289750072857579645, 7.935384661251128756487087682079