# Properties

 Degree $2$ Conductor $700$ Sign $0.502 - 0.864i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 + (0.366 − 1.36i)2-s + (−1.5 − 0.866i)3-s + (−1.73 − i)4-s + (−1.73 + 1.73i)6-s + (−2 − 1.73i)7-s + (−2 + 1.99i)8-s + (0.866 + 0.5i)11-s + (1.73 + 3i)12-s − 3.46·13-s + (−3.09 + 2.09i)14-s + (1.99 + 3.46i)16-s + (0.866 − 1.5i)17-s + (2.59 + 4.5i)19-s + (1.50 + 4.33i)21-s + (1 − 0.999i)22-s + (−0.5 − 0.866i)23-s + ⋯
 L(s)  = 1 + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)3-s + (−0.866 − 0.5i)4-s + (−0.707 + 0.707i)6-s + (−0.755 − 0.654i)7-s + (−0.707 + 0.707i)8-s + (0.261 + 0.150i)11-s + (0.499 + 0.866i)12-s − 0.960·13-s + (−0.827 + 0.560i)14-s + (0.499 + 0.866i)16-s + (0.210 − 0.363i)17-s + (0.596 + 1.03i)19-s + (0.327 + 0.944i)21-s + (0.213 − 0.213i)22-s + (−0.104 − 0.180i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$700$$    =    $$2^{2} \cdot 5^{2} \cdot 7$$ Sign: $0.502 - 0.864i$ Motivic weight: $$1$$ Character: $\chi_{700} (199, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$1$$ Selberg data: $$(2,\ 700,\ (\ :1/2),\ 0.502 - 0.864i)$$

## 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$$F_p(T)$
bad2 $$1 + (-0.366 + 1.36i)T$$
5 $$1$$
7 $$1 + (2 + 1.73i)T$$
good3 $$1 + (1.5 + 0.866i)T + (1.5 + 2.59i)T^{2}$$
11 $$1 + (-0.866 - 0.5i)T + (5.5 + 9.52i)T^{2}$$
13 $$1 + 3.46T + 13T^{2}$$
17 $$1 + (-0.866 + 1.5i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-2.59 - 4.5i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + 4T + 29T^{2}$$
31 $$1 + (0.866 - 1.5i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (-2.59 + 1.5i)T + (18.5 - 32.0i)T^{2}$$
41 $$1 - 3.46iT - 41T^{2}$$
43 $$1 - 2T + 43T^{2}$$
47 $$1 + (7.5 - 4.33i)T + (23.5 - 40.7i)T^{2}$$
53 $$1 + (0.866 + 0.5i)T + (26.5 + 45.8i)T^{2}$$
59 $$1 + (-2.59 + 4.5i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2}$$
67 $$1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 14iT - 71T^{2}$$
73 $$1 + (-4.33 + 7.5i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (7.79 - 4.5i)T + (39.5 - 68.4i)T^{2}$$
83 $$1 - 13.8iT - 83T^{2}$$
89 $$1 + (13.5 - 7.79i)T + (44.5 - 77.0i)T^{2}$$
97 $$1 + 17.3T + 97T^{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

−9.775858298332276670377244079219, −9.404063633507386959706419921209, −7.917279309222902455252546946437, −6.91643814560194720328877919890, −6.00802259076958612267629136481, −5.14450797898659285188970422895, −3.98288927177179315572824353925, −2.95585148792669214423101273942, −1.35875752573773810669244344562, 0, 2.82535537886469655401796563754, 4.11056799359014177845823238096, 5.16839598387022676127077856254, 5.69160011151600175782804437617, 6.60719548535132736608479386710, 7.47457579605821980622061543019, 8.578142902864523547717124533244, 9.482950740475853154850262529895, 10.03590938697895814323944038614