# Properties

 Label 2-105-105.104-c3-0-22 Degree $2$ Conductor $105$ Sign $0.0658 + 0.997i$ Analytic cond. $6.19520$ Root an. cond. $2.48901$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.24·2-s + (1.06 + 5.08i)3-s − 2.96·4-s + (−8.51 + 7.24i)5-s + (−2.39 − 11.4i)6-s + (−12.2 − 13.8i)7-s + 24.6·8-s + (−24.7 + 10.8i)9-s + (19.1 − 16.2i)10-s − 25.7i·11-s + (−3.16 − 15.0i)12-s + 68.2·13-s + (27.5 + 31.1i)14-s + (−45.9 − 35.5i)15-s − 31.5·16-s + 30.6i·17-s + ⋯
 L(s)  = 1 − 0.793·2-s + (0.205 + 0.978i)3-s − 0.370·4-s + (−0.761 + 0.648i)5-s + (−0.162 − 0.776i)6-s + (−0.662 − 0.748i)7-s + 1.08·8-s + (−0.915 + 0.401i)9-s + (0.604 − 0.514i)10-s − 0.705i·11-s + (−0.0760 − 0.362i)12-s + 1.45·13-s + (0.526 + 0.594i)14-s + (−0.790 − 0.612i)15-s − 0.492·16-s + 0.437i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0658 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0658 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$105$$    =    $$3 \cdot 5 \cdot 7$$ Sign: $0.0658 + 0.997i$ Analytic conductor: $$6.19520$$ Root analytic conductor: $$2.48901$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{105} (104, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 105,\ (\ :3/2),\ 0.0658 + 0.997i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.206367 - 0.193191i$$ $$L(\frac12)$$ $$\approx$$ $$0.206367 - 0.193191i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-1.06 - 5.08i)T$$
5 $$1 + (8.51 - 7.24i)T$$
7 $$1 + (12.2 + 13.8i)T$$
good2 $$1 + 2.24T + 8T^{2}$$
11 $$1 + 25.7iT - 1.33e3T^{2}$$
13 $$1 - 68.2T + 2.19e3T^{2}$$
17 $$1 - 30.6iT - 4.91e3T^{2}$$
19 $$1 + 109. iT - 6.85e3T^{2}$$
23 $$1 + 152.T + 1.21e4T^{2}$$
29 $$1 + 191. iT - 2.43e4T^{2}$$
31 $$1 + 16.4iT - 2.97e4T^{2}$$
37 $$1 - 81.9iT - 5.06e4T^{2}$$
41 $$1 + 372.T + 6.89e4T^{2}$$
43 $$1 - 192. iT - 7.95e4T^{2}$$
47 $$1 + 0.366iT - 1.03e5T^{2}$$
53 $$1 - 5.95T + 1.48e5T^{2}$$
59 $$1 - 198.T + 2.05e5T^{2}$$
61 $$1 + 83.5iT - 2.26e5T^{2}$$
67 $$1 + 1.08e3iT - 3.00e5T^{2}$$
71 $$1 + 773. iT - 3.57e5T^{2}$$
73 $$1 + 448.T + 3.89e5T^{2}$$
79 $$1 + 1.27e3T + 4.93e5T^{2}$$
83 $$1 - 429. iT - 5.71e5T^{2}$$
89 $$1 + 5.29T + 7.04e5T^{2}$$
97 $$1 - 435.T + 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

−13.42566588052436120258093805008, −11.41173543469952145958636086155, −10.67708006761820712194288938802, −9.872375603291558275085051385323, −8.689232275360081658498019363291, −7.899941088126371927059810788698, −6.30208986379165146959652043030, −4.30797448049193326439737633023, −3.42538886038947228151673049278, −0.21607152127359058306803990742, 1.48970580128288132586034577057, 3.76079823473720293595491074884, 5.64624062037845153782838492788, 7.16319555457547866811757361560, 8.352908734388586034743896974783, 8.788266252105555696381740683663, 10.06235616436935404591717406705, 11.65868406565112284181143987077, 12.51948514830742715615852434694, 13.28771576996359969529335832569