# Properties

 Label 2-300-15.2-c1-0-3 Degree $2$ Conductor $300$ Sign $0.945 + 0.326i$ Analytic cond. $2.39551$ Root an. cond. $1.54774$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Learn more

## Dirichlet series

 L(s)  = 1 + (1.22 − 1.22i)3-s + (3.67 + 3.67i)7-s − 2.99i·9-s + (−1.22 + 1.22i)13-s − 7i·19-s + 9·21-s + (−3.67 − 3.67i)27-s − 11·31-s + (4.89 + 4.89i)37-s + 2.99i·39-s + (−1.22 + 1.22i)43-s + 20i·49-s + (−8.57 − 8.57i)57-s − 61-s + (11.0 − 11.0i)63-s + ⋯
 L(s)  = 1 + (0.707 − 0.707i)3-s + (1.38 + 1.38i)7-s − 0.999i·9-s + (−0.339 + 0.339i)13-s − 1.60i·19-s + 1.96·21-s + (−0.707 − 0.707i)27-s − 1.97·31-s + (0.805 + 0.805i)37-s + 0.480i·39-s + (−0.186 + 0.186i)43-s + 2.85i·49-s + (−1.13 − 1.13i)57-s − 0.128·61-s + (1.38 − 1.38i)63-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$300$$    =    $$2^{2} \cdot 3 \cdot 5^{2}$$ Sign: $0.945 + 0.326i$ Analytic conductor: $$2.39551$$ Root analytic conductor: $$1.54774$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{300} (257, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 300,\ (\ :1/2),\ 0.945 + 0.326i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.73555 - 0.291225i$$ $$L(\frac12)$$ $$\approx$$ $$1.73555 - 0.291225i$$ $$L(\frac{3}{2})$$ 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 + (-1.22 + 1.22i)T$$
5 $$1$$
good7 $$1 + (-3.67 - 3.67i)T + 7iT^{2}$$
11 $$1 - 11T^{2}$$
13 $$1 + (1.22 - 1.22i)T - 13iT^{2}$$
17 $$1 - 17iT^{2}$$
19 $$1 + 7iT - 19T^{2}$$
23 $$1 + 23iT^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 + 11T + 31T^{2}$$
37 $$1 + (-4.89 - 4.89i)T + 37iT^{2}$$
41 $$1 - 41T^{2}$$
43 $$1 + (1.22 - 1.22i)T - 43iT^{2}$$
47 $$1 - 47iT^{2}$$
53 $$1 + 53iT^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 + T + 61T^{2}$$
67 $$1 + (8.57 + 8.57i)T + 67iT^{2}$$
71 $$1 - 71T^{2}$$
73 $$1 + (9.79 - 9.79i)T - 73iT^{2}$$
79 $$1 - 4iT - 79T^{2}$$
83 $$1 + 83iT^{2}$$
89 $$1 + 89T^{2}$$
97 $$1 + (-3.67 - 3.67i)T + 97iT^{2}$$
show more
show less
$$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

−11.72877412032451963904592078077, −11.09738498905289524772231121769, −9.377872132847531494493509109494, −8.822555355197016147019460866317, −7.953120423714196314595464752976, −7.00924955764891364215982017054, −5.71177491798444827396276226959, −4.60886312772752862324915921247, −2.80015991950373825266521744282, −1.78628838782571669735172043363, 1.77973688557852886147883827335, 3.61653648692917148728047209142, 4.44020035648509998984483148507, 5.52482853605190452122363885586, 7.43842888431618224300548340974, 7.84148658730117904538530503067, 8.915701980423007964693690494091, 10.13113842523790959479490727842, 10.64422132286676405041391494598, 11.52030785592762076557666464599