# Properties

 Label 2-105-105.104-c3-0-26 Degree $2$ Conductor $105$ Sign $0.781 + 0.623i$ 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 − 1.22·2-s + (4.91 − 1.67i)3-s − 6.50·4-s + (9.89 − 5.21i)5-s + (−6.01 + 2.04i)6-s + (1.55 + 18.4i)7-s + 17.7·8-s + (21.4 − 16.4i)9-s + (−12.0 + 6.37i)10-s − 54.5i·11-s + (−32.0 + 10.8i)12-s + 24.4·13-s + (−1.90 − 22.5i)14-s + (39.9 − 42.1i)15-s + 30.3·16-s + 36.0i·17-s + ⋯
 L(s)  = 1 − 0.432·2-s + (0.946 − 0.322i)3-s − 0.813·4-s + (0.884 − 0.466i)5-s + (−0.409 + 0.139i)6-s + (0.0841 + 0.996i)7-s + 0.783·8-s + (0.792 − 0.609i)9-s + (−0.382 + 0.201i)10-s − 1.49i·11-s + (−0.769 + 0.261i)12-s + 0.522·13-s + (−0.0363 − 0.430i)14-s + (0.687 − 0.726i)15-s + 0.474·16-s + 0.513i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$105$$    =    $$3 \cdot 5 \cdot 7$$ Sign: $0.781 + 0.623i$ 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: $$(2,\ 105,\ (\ :3/2),\ 0.781 + 0.623i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.61831 - 0.566634i$$ $$L(\frac12)$$ $$\approx$$ $$1.61831 - 0.566634i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-4.91 + 1.67i)T$$
5 $$1 + (-9.89 + 5.21i)T$$
7 $$1 + (-1.55 - 18.4i)T$$
good2 $$1 + 1.22T + 8T^{2}$$
11 $$1 + 54.5iT - 1.33e3T^{2}$$
13 $$1 - 24.4T + 2.19e3T^{2}$$
17 $$1 - 36.0iT - 4.91e3T^{2}$$
19 $$1 + 99.5iT - 6.85e3T^{2}$$
23 $$1 - 97.3T + 1.21e4T^{2}$$
29 $$1 + 14.1iT - 2.43e4T^{2}$$
31 $$1 - 186. iT - 2.97e4T^{2}$$
37 $$1 - 199. iT - 5.06e4T^{2}$$
41 $$1 + 313.T + 6.89e4T^{2}$$
43 $$1 + 30.0iT - 7.95e4T^{2}$$
47 $$1 - 514. iT - 1.03e5T^{2}$$
53 $$1 + 451.T + 1.48e5T^{2}$$
59 $$1 + 355.T + 2.05e5T^{2}$$
61 $$1 - 656. iT - 2.26e5T^{2}$$
67 $$1 + 73.7iT - 3.00e5T^{2}$$
71 $$1 + 611. iT - 3.57e5T^{2}$$
73 $$1 - 424.T + 3.89e5T^{2}$$
79 $$1 - 814.T + 4.93e5T^{2}$$
83 $$1 - 585. iT - 5.71e5T^{2}$$
89 $$1 - 732.T + 7.04e5T^{2}$$
97 $$1 + 272.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

−13.42596526408557702791917929088, −12.49491125649768954576150663551, −10.82171050153827116273146280394, −9.422107236235323231785142055773, −8.813404403397637143442464135887, −8.246327608336585309247309575656, −6.32986459985884526298470578160, −4.97575871167518981227685825006, −3.07330623881337971534832119007, −1.25433630125440740354803222747, 1.72265005840392622990174503772, 3.71570896726810436247453466698, 4.94517334004510804562314437463, 7.01211650695401529645961783531, 7.981586072065595982608853129708, 9.346112485630464783520133847915, 9.923857034439032355406905532466, 10.69584098849344428505936190329, 12.80234100938101621664638255138, 13.58411627939539874187977484241