# Properties

 Label 2-105-5.4-c3-0-2 Degree $2$ Conductor $105$ Sign $0.861 - 0.508i$ Analytic cond. $6.19520$ Root an. cond. $2.48901$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 4.88i·2-s + 3i·3-s − 15.9·4-s + (−9.63 + 5.67i)5-s + 14.6·6-s + 7i·7-s + 38.6i·8-s − 9·9-s + (27.7 + 47.0i)10-s + 54.9·11-s − 47.7i·12-s + 49.7i·13-s + 34.2·14-s + (−17.0 − 28.8i)15-s + 61.7·16-s + 133. i·17-s + ⋯
 L(s)  = 1 − 1.72i·2-s + 0.577i·3-s − 1.98·4-s + (−0.861 + 0.508i)5-s + 0.998·6-s + 0.377i·7-s + 1.70i·8-s − 0.333·9-s + (0.878 + 1.48i)10-s + 1.50·11-s − 1.14i·12-s + 1.06i·13-s + 0.653·14-s + (−0.293 − 0.497i)15-s + 0.964·16-s + 1.90i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$105$$    =    $$3 \cdot 5 \cdot 7$$ Sign: $0.861 - 0.508i$ Analytic conductor: $$6.19520$$ Root analytic conductor: $$2.48901$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{105} (64, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 105,\ (\ :3/2),\ 0.861 - 0.508i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.755842 + 0.206288i$$ $$L(\frac12)$$ $$\approx$$ $$0.755842 + 0.206288i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 3iT$$
5 $$1 + (9.63 - 5.67i)T$$
7 $$1 - 7iT$$
good2 $$1 + 4.88iT - 8T^{2}$$
11 $$1 - 54.9T + 1.33e3T^{2}$$
13 $$1 - 49.7iT - 2.19e3T^{2}$$
17 $$1 - 133. iT - 4.91e3T^{2}$$
19 $$1 + 138.T + 6.85e3T^{2}$$
23 $$1 + 7.32iT - 1.21e4T^{2}$$
29 $$1 + 87.2T + 2.43e4T^{2}$$
31 $$1 + 209.T + 2.97e4T^{2}$$
37 $$1 + 67.9iT - 5.06e4T^{2}$$
41 $$1 - 77.6T + 6.89e4T^{2}$$
43 $$1 - 197. iT - 7.95e4T^{2}$$
47 $$1 - 4.97iT - 1.03e5T^{2}$$
53 $$1 - 53.0iT - 1.48e5T^{2}$$
59 $$1 - 683.T + 2.05e5T^{2}$$
61 $$1 + 26.8T + 2.26e5T^{2}$$
67 $$1 - 149. iT - 3.00e5T^{2}$$
71 $$1 - 6.15T + 3.57e5T^{2}$$
73 $$1 + 294. iT - 3.89e5T^{2}$$
79 $$1 - 938.T + 4.93e5T^{2}$$
83 $$1 + 784. iT - 5.71e5T^{2}$$
89 $$1 + 275.T + 7.04e5T^{2}$$
97 $$1 - 1.16e3iT - 9.12e5T^{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

−12.88700374079946699487414139962, −12.03772515401801868329612376679, −11.20932378825912725843916749142, −10.57109110606400198242669912792, −9.275927268385660273364778970614, −8.524443798674431410728837745220, −6.44236979367788911178769130853, −4.19181882194962807505477648783, −3.77129916583061970201418224630, −1.93752015487799232672000412063, 0.44928666203797719394675372730, 3.99974671043869865632450468572, 5.30095993833632765420276781059, 6.69644002191792149335098483460, 7.42330824102590640755743711813, 8.442808553472678563399484452737, 9.304928023744878854175232108392, 11.27743264105076338988419045449, 12.48698607219363729038366796433, 13.45599382062213503484712799753