# Properties

 Label 2-1200-60.23-c0-0-0 Degree $2$ Conductor $1200$ Sign $0.793 - 0.608i$ Analytic cond. $0.598878$ Root an. cond. $0.773872$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.707 + 0.707i)3-s + (−0.707 − 0.707i)7-s − 1.00i·9-s + (1.22 + 1.22i)13-s + 1.73·19-s + 1.00·21-s + (0.707 + 0.707i)27-s + 1.73i·31-s − 1.73·39-s + (0.707 − 0.707i)43-s + (−1.22 + 1.22i)57-s − 61-s + (−0.707 + 0.707i)63-s + (0.707 + 0.707i)67-s − 1.00·81-s + ⋯
 L(s)  = 1 + (−0.707 + 0.707i)3-s + (−0.707 − 0.707i)7-s − 1.00i·9-s + (1.22 + 1.22i)13-s + 1.73·19-s + 1.00·21-s + (0.707 + 0.707i)27-s + 1.73i·31-s − 1.73·39-s + (0.707 − 0.707i)43-s + (−1.22 + 1.22i)57-s − 61-s + (−0.707 + 0.707i)63-s + (0.707 + 0.707i)67-s − 1.00·81-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1200$$    =    $$2^{4} \cdot 3 \cdot 5^{2}$$ Sign: $0.793 - 0.608i$ Analytic conductor: $$0.598878$$ Root analytic conductor: $$0.773872$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1200} (143, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1200,\ (\ :0),\ 0.793 - 0.608i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8291653577$$ $$L(\frac12)$$ $$\approx$$ $$0.8291653577$$ $$L(1)$$ 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 + (0.707 - 0.707i)T$$
5 $$1$$
good7 $$1 + (0.707 + 0.707i)T + iT^{2}$$
11 $$1 + T^{2}$$
13 $$1 + (-1.22 - 1.22i)T + iT^{2}$$
17 $$1 + iT^{2}$$
19 $$1 - 1.73T + T^{2}$$
23 $$1 + iT^{2}$$
29 $$1 + T^{2}$$
31 $$1 - 1.73iT - T^{2}$$
37 $$1 - iT^{2}$$
41 $$1 - T^{2}$$
43 $$1 + (-0.707 + 0.707i)T - iT^{2}$$
47 $$1 - iT^{2}$$
53 $$1 - iT^{2}$$
59 $$1 - T^{2}$$
61 $$1 + T + T^{2}$$
67 $$1 + (-0.707 - 0.707i)T + iT^{2}$$
71 $$1 + T^{2}$$
73 $$1 + iT^{2}$$
79 $$1 + T^{2}$$
83 $$1 + iT^{2}$$
89 $$1 + T^{2}$$
97 $$1 + (-1.22 + 1.22i)T - iT^{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

−10.08507196645866072445573421240, −9.325262658929576262135558019633, −8.696971499392248197459982456699, −7.29534516278457919504371112267, −6.66738458061285020198690160939, −5.84715618838192820591675114199, −4.88567892708054104560994094378, −3.89028868211495220529802022498, −3.26709994203518284727793812159, −1.22464184173182622059227492847, 1.02774388645613217780550038082, 2.56986317391321156244771585004, 3.55209047825886709964711488501, 5.04261065647822190147318681762, 5.89679762857214034794512502421, 6.22961836130339649427774596142, 7.48673346989188746276426712887, 8.002498012967468546610284351927, 9.082636131720859082930885620306, 9.877385643951298330637270979361