# Properties

 Label 2-124-124.67-c0-0-1 Degree $2$ Conductor $124$ Sign $-0.390 + 0.920i$ Analytic cond. $0.0618840$ Root an. cond. $0.248765$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s + (−0.866 − 0.5i)3-s − 4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (0.866 + 0.5i)7-s + i·8-s + (−0.866 + 0.5i)10-s + (0.866 − 0.5i)11-s + (0.866 + 0.5i)12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + 0.999i·15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯
 L(s)  = 1 − i·2-s + (−0.866 − 0.5i)3-s − 4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (0.866 + 0.5i)7-s + i·8-s + (−0.866 + 0.5i)10-s + (0.866 − 0.5i)11-s + (0.866 + 0.5i)12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + 0.999i·15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$124$$    =    $$2^{2} \cdot 31$$ Sign: $-0.390 + 0.920i$ Analytic conductor: $$0.0618840$$ Root analytic conductor: $$0.248765$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{124} (67, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 124,\ (\ :0),\ -0.390 + 0.920i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.4496221474$$ $$L(\frac12)$$ $$\approx$$ $$0.4496221474$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
31 $$1 - iT$$
good3 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
5 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
7 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
11 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
13 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
17 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
19 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
23 $$1 - T^{2}$$
29 $$1 + T^{2}$$
37 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}$$
41 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
43 $$1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
59 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
61 $$1 + T^{2}$$
67 $$1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}$$
71 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
73 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
79 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
83 $$1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}$$
89 $$1 + T^{2}$$
97 $$1 + 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

−12.91837495169102241744881554199, −11.99553861310626440190086689809, −11.61865458042174095040592607421, −10.69380714326108650488253231074, −8.768357939163126364910945396664, −8.622440126044797827118706350253, −6.51577319627012882774980448639, −5.15770407288810986410585864163, −4.01588990366557451481704433847, −1.52908883261400833847866302720, 3.94693333798037946909518849036, 5.01178891049608934019955746078, 6.30474776983992920335379169545, 7.35518884853854989998619678137, 8.387361065421477009127377174460, 9.935290020569122213739987438421, 10.90414473034916737584421224431, 11.67379482032757640573734427078, 13.18153866567090037113711101746, 14.38064540906110877033278196375