# Properties

 Label 2-105-5.4-c3-0-4 Degree $2$ Conductor $105$ Sign $0.158 - 0.987i$ 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 − 0.428i·2-s + 3i·3-s + 7.81·4-s + (−1.76 + 11.0i)5-s + 1.28·6-s + 7i·7-s − 6.77i·8-s − 9·9-s + (4.72 + 0.757i)10-s − 27.4·11-s + 23.4i·12-s + 46.5i·13-s + 2.99·14-s + (−33.1 − 5.30i)15-s + 59.6·16-s + 5.20i·17-s + ⋯
 L(s)  = 1 − 0.151i·2-s + 0.577i·3-s + 0.977·4-s + (−0.158 + 0.987i)5-s + 0.0874·6-s + 0.377i·7-s − 0.299i·8-s − 0.333·9-s + (0.149 + 0.0239i)10-s − 0.753·11-s + 0.564i·12-s + 0.993i·13-s + 0.0572·14-s + (−0.570 − 0.0913i)15-s + 0.931·16-s + 0.0742i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$105$$    =    $$3 \cdot 5 \cdot 7$$ Sign: $0.158 - 0.987i$ 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: $$(2,\ 105,\ (\ :3/2),\ 0.158 - 0.987i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.33411 + 1.13737i$$ $$L(\frac12)$$ $$\approx$$ $$1.33411 + 1.13737i$$ $$L(\frac{5}{2})$$ 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 + (1.76 - 11.0i)T$$
7 $$1 - 7iT$$
good2 $$1 + 0.428iT - 8T^{2}$$
11 $$1 + 27.4T + 1.33e3T^{2}$$
13 $$1 - 46.5iT - 2.19e3T^{2}$$
17 $$1 - 5.20iT - 4.91e3T^{2}$$
19 $$1 - 91.0T + 6.85e3T^{2}$$
23 $$1 - 111. iT - 1.21e4T^{2}$$
29 $$1 + 0.0763T + 2.43e4T^{2}$$
31 $$1 - 201.T + 2.97e4T^{2}$$
37 $$1 + 312. iT - 5.06e4T^{2}$$
41 $$1 - 102.T + 6.89e4T^{2}$$
43 $$1 + 257. iT - 7.95e4T^{2}$$
47 $$1 + 350. iT - 1.03e5T^{2}$$
53 $$1 + 196. iT - 1.48e5T^{2}$$
59 $$1 + 881.T + 2.05e5T^{2}$$
61 $$1 - 737.T + 2.26e5T^{2}$$
67 $$1 - 365. iT - 3.00e5T^{2}$$
71 $$1 - 1.11e3T + 3.57e5T^{2}$$
73 $$1 + 261. iT - 3.89e5T^{2}$$
79 $$1 + 273.T + 4.93e5T^{2}$$
83 $$1 - 87.1iT - 5.71e5T^{2}$$
89 $$1 - 1.09e3T + 7.04e5T^{2}$$
97 $$1 - 228. iT - 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

−13.69848464714311473104675022402, −12.05469494883976185694999577586, −11.36345180390108925763177355713, −10.47981240725889895473708313187, −9.499475118200740161686517061087, −7.82522180223221112132653262826, −6.78950790966374679914279276152, −5.52797727009820692301037337687, −3.58427089350322792791823089277, −2.34324822840924792785866273889, 1.01861256683466620145631022893, 2.87180820998379020100398082014, 5.00718985857788251436213541895, 6.23923352954947796229789468733, 7.63260051473435191132272506511, 8.221858903409402363274938119492, 9.918430751862060733641999654867, 11.07807238437069614113528286092, 12.15410419008600437938056181899, 12.88577672339804314219225100269