# Properties

 Label 1-105-105.17-r1-0-0 Degree $1$ Conductor $105$ Sign $-0.156 + 0.987i$ Analytic cond. $11.2838$ Root an. cond. $11.2838$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + i·8-s + (0.5 + 0.866i)11-s + i·13-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + i·22-s + (−0.866 − 0.5i)23-s + (−0.5 + 0.866i)26-s + 29-s + (0.5 + 0.866i)31-s + (−0.866 + 0.5i)32-s + 34-s + ⋯
 L(s)  = 1 + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + i·8-s + (0.5 + 0.866i)11-s + i·13-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + i·22-s + (−0.866 − 0.5i)23-s + (−0.5 + 0.866i)26-s + 29-s + (0.5 + 0.866i)31-s + (−0.866 + 0.5i)32-s + 34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$105$$    =    $$3 \cdot 5 \cdot 7$$ Sign: $-0.156 + 0.987i$ Analytic conductor: $$11.2838$$ Root analytic conductor: $$11.2838$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{105} (17, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 105,\ (1:\ ),\ -0.156 + 0.987i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.811389767 + 2.120105693i$$ $$L(\frac12)$$ $$\approx$$ $$1.811389767 + 2.120105693i$$ $$L(1)$$ $$\approx$$ $$1.558988302 + 0.8671390436i$$ $$L(1)$$ $$\approx$$ $$1.558988302 + 0.8671390436i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
7 $$1$$
good2 $$1 + (0.866 + 0.5i)T$$
11 $$1 + (0.5 + 0.866i)T$$
13 $$1 + iT$$
17 $$1 + (0.866 - 0.5i)T$$
19 $$1 + (-0.5 + 0.866i)T$$
23 $$1 + (-0.866 - 0.5i)T$$
29 $$1 + T$$
31 $$1 + (0.5 + 0.866i)T$$
37 $$1 + (-0.866 - 0.5i)T$$
41 $$1 + T$$
43 $$1 - iT$$
47 $$1 + (-0.866 - 0.5i)T$$
53 $$1 + (0.866 - 0.5i)T$$
59 $$1 + (0.5 + 0.866i)T$$
61 $$1 + (0.5 - 0.866i)T$$
67 $$1 + (0.866 - 0.5i)T$$
71 $$1 - T$$
73 $$1 + (0.866 - 0.5i)T$$
79 $$1 + (0.5 - 0.866i)T$$
83 $$1 - iT$$
89 $$1 + (0.5 - 0.866i)T$$
97 $$1 - iT$$
show less
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−29.65162769126381317801274013190, −28.1711248653568268107540459891, −27.51279374403017242299167233018, −25.938285959553782568027316106607, −24.77700019899877822262600810081, −23.901151945483155487532745908009, −22.85279667960204528940062901071, −21.87356702915152994372663677151, −21.03774849337250959819659031050, −19.80112472080749755482284656710, −19.08716537654744831637364527077, −17.61784432691752959817640987837, −16.18914085650522333433797709483, −15.119154594873304106534882632861, −14.05444977457702051226449324128, −13.04730583625022113026095806874, −11.940327208715621195576407937241, −10.869241114657138087503112804798, −9.77315740806023969448895588, −8.13770839572759376426490657176, −6.45733475462387093109276529267, −5.43547563572545142456145900971, −3.962061405869019912897533012534, −2.763489543264713663393180448041, −0.97152758335965067345692074834, 2.034247183860731884092781772548, 3.73164202577169353113149144993, 4.83431089499032463264806659654, 6.26738410863518089795006908046, 7.27801141727382561322340482392, 8.613216779714191651174558326139, 10.16653743773725181359665410326, 11.82649685111012507425664641583, 12.44002784875433066084308874638, 13.98144223835120123844049983909, 14.579262474619503434295432482655, 15.922607923618600043139464964, 16.784826766044178160207265917338, 17.93747792327814077191974012194, 19.42611725196674377781380428766, 20.68541964978492134831108076776, 21.50320556327192892116736238898, 22.71825707983812011733009545939, 23.397069710082722830759158485108, 24.587040105164494436134299761983, 25.43372122304953932068851038632, 26.37627833044666502072135912074, 27.62914842289857164076855848802, 28.93765441335301439070310883678, 29.99254568077376053088559787444