# Properties

 Label 2-105-1.1-c3-0-9 Degree $2$ Conductor $105$ Sign $1$ Analytic cond. $6.19520$ Root an. cond. $2.48901$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 4.53·2-s + 3·3-s + 12.5·4-s + 5·5-s + 13.5·6-s − 7·7-s + 20.5·8-s + 9·9-s + 22.6·10-s − 19.0·11-s + 37.5·12-s − 2.93·13-s − 31.7·14-s + 15·15-s − 7.21·16-s − 6.49·17-s + 40.7·18-s − 5.43·19-s + 62.6·20-s − 21·21-s − 86.3·22-s + 49.3·23-s + 61.5·24-s + 25·25-s − 13.3·26-s + 27·27-s − 87.7·28-s + ⋯
 L(s)  = 1 + 1.60·2-s + 0.577·3-s + 1.56·4-s + 0.447·5-s + 0.924·6-s − 0.377·7-s + 0.907·8-s + 0.333·9-s + 0.716·10-s − 0.522·11-s + 0.904·12-s − 0.0626·13-s − 0.605·14-s + 0.258·15-s − 0.112·16-s − 0.0927·17-s + 0.533·18-s − 0.0656·19-s + 0.700·20-s − 0.218·21-s − 0.837·22-s + 0.447·23-s + 0.523·24-s + 0.200·25-s − 0.100·26-s + 0.192·27-s − 0.592·28-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$105$$    =    $$3 \cdot 5 \cdot 7$$ Sign: $1$ Analytic conductor: $$6.19520$$ Root analytic conductor: $$2.48901$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 105,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$4.178010652$$ $$L(\frac12)$$ $$\approx$$ $$4.178010652$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 3T$$
5 $$1 - 5T$$
7 $$1 + 7T$$
good2 $$1 - 4.53T + 8T^{2}$$
11 $$1 + 19.0T + 1.33e3T^{2}$$
13 $$1 + 2.93T + 2.19e3T^{2}$$
17 $$1 + 6.49T + 4.91e3T^{2}$$
19 $$1 + 5.43T + 6.85e3T^{2}$$
23 $$1 - 49.3T + 1.21e4T^{2}$$
29 $$1 + 291.T + 2.43e4T^{2}$$
31 $$1 - 244.T + 2.97e4T^{2}$$
37 $$1 + 193.T + 5.06e4T^{2}$$
41 $$1 - 315.T + 6.89e4T^{2}$$
43 $$1 + 300.T + 7.95e4T^{2}$$
47 $$1 - 86.5T + 1.03e5T^{2}$$
53 $$1 - 509.T + 1.48e5T^{2}$$
59 $$1 + 83.3T + 2.05e5T^{2}$$
61 $$1 + 5.25T + 2.26e5T^{2}$$
67 $$1 - 205.T + 3.00e5T^{2}$$
71 $$1 - 1.00e3T + 3.57e5T^{2}$$
73 $$1 + 1.00e3T + 3.89e5T^{2}$$
79 $$1 + 863.T + 4.93e5T^{2}$$
83 $$1 - 1.33e3T + 5.71e5T^{2}$$
89 $$1 - 326.T + 7.04e5T^{2}$$
97 $$1 - 1.52e3T + 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

−13.30346175272847329227780832686, −12.73831602913954178485104586965, −11.53088546756880426510015652679, −10.24689496600514792997960725745, −8.959543556742975218076338319296, −7.35197583697436006371302737442, −6.13911297908769492799448079281, −4.97877979516232912120258346709, −3.60040878612527266076080746053, −2.36836063550997595589839875669, 2.36836063550997595589839875669, 3.60040878612527266076080746053, 4.97877979516232912120258346709, 6.13911297908769492799448079281, 7.35197583697436006371302737442, 8.959543556742975218076338319296, 10.24689496600514792997960725745, 11.53088546756880426510015652679, 12.73831602913954178485104586965, 13.30346175272847329227780832686