# Properties

 Label 2-300-20.19-c2-0-10 Degree $2$ Conductor $300$ Sign $-0.591 - 0.806i$ Analytic cond. $8.17440$ Root an. cond. $2.85909$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.169 + 1.99i)2-s − 1.73·3-s + (−3.94 + 0.675i)4-s + (−0.293 − 3.45i)6-s + 12.3·7-s + (−2.01 − 7.74i)8-s + 2.99·9-s + 11.0i·11-s + (6.82 − 1.16i)12-s + 2.82i·13-s + (2.10 + 24.7i)14-s + (15.0 − 5.32i)16-s − 6.52i·17-s + (0.508 + 5.97i)18-s + 27.9i·19-s + ⋯
 L(s)  = 1 + (0.0847 + 0.996i)2-s − 0.577·3-s + (−0.985 + 0.168i)4-s + (−0.0489 − 0.575i)6-s + 1.77·7-s + (−0.251 − 0.967i)8-s + 0.333·9-s + 1.00i·11-s + (0.569 − 0.0974i)12-s + 0.216i·13-s + (0.150 + 1.76i)14-s + (0.942 − 0.332i)16-s − 0.383i·17-s + (0.0282 + 0.332i)18-s + 1.47i·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$300$$    =    $$2^{2} \cdot 3 \cdot 5^{2}$$ Sign: $-0.591 - 0.806i$ Analytic conductor: $$8.17440$$ Root analytic conductor: $$2.85909$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{300} (199, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 300,\ (\ :1),\ -0.591 - 0.806i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.626178 + 1.23654i$$ $$L(\frac12)$$ $$\approx$$ $$0.626178 + 1.23654i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.169 - 1.99i)T$$
3 $$1 + 1.73T$$
5 $$1$$
good7 $$1 - 12.3T + 49T^{2}$$
11 $$1 - 11.0iT - 121T^{2}$$
13 $$1 - 2.82iT - 169T^{2}$$
17 $$1 + 6.52iT - 289T^{2}$$
19 $$1 - 27.9iT - 361T^{2}$$
23 $$1 - 7.90T + 529T^{2}$$
29 $$1 + 50.7T + 841T^{2}$$
31 $$1 - 36.3iT - 961T^{2}$$
37 $$1 - 18.9iT - 1.36e3T^{2}$$
41 $$1 - 5.30T + 1.68e3T^{2}$$
43 $$1 - 45.5T + 1.84e3T^{2}$$
47 $$1 - 11.7T + 2.20e3T^{2}$$
53 $$1 - 41.1iT - 2.80e3T^{2}$$
59 $$1 + 10.7iT - 3.48e3T^{2}$$
61 $$1 - 56.1T + 3.72e3T^{2}$$
67 $$1 + 16.1T + 4.48e3T^{2}$$
71 $$1 - 66.1iT - 5.04e3T^{2}$$
73 $$1 - 15.6iT - 5.32e3T^{2}$$
79 $$1 + 123. iT - 6.24e3T^{2}$$
83 $$1 - 99.6T + 6.88e3T^{2}$$
89 $$1 + 101.T + 7.92e3T^{2}$$
97 $$1 + 127. iT - 9.40e3T^{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

−11.95467109397428469919659262172, −10.94404891645348638231869146573, −9.899707315171950341859908164584, −8.782555266835479178432428812500, −7.75089271389876864358946755068, −7.15828093925966569140246887645, −5.75961716805033972305128143721, −4.96737637767451009430120086997, −4.10289474983015202211536535562, −1.55916413286521470525823595412, 0.797377112131599402338939739939, 2.22160864579171495833824578182, 3.93133387534497842607880263694, 4.99579307528422239074732499027, 5.76463275806659948465636421671, 7.54432469770334698440757552082, 8.497238948182549365056931517294, 9.395080273655115243729291972493, 10.92762469838908193467538632281, 11.02369505742123600392842620295