# Properties

 Degree $2$ Conductor $8034$ Sign $1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $0$

# Learn more about

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s + 4-s − 1.52·5-s + 6-s − 3.40·7-s − 8-s + 9-s + 1.52·10-s + 2.89·11-s − 12-s + 13-s + 3.40·14-s + 1.52·15-s + 16-s + 3.63·17-s − 18-s + 8.33·19-s − 1.52·20-s + 3.40·21-s − 2.89·22-s − 8.49·23-s + 24-s − 2.68·25-s − 26-s − 27-s − 3.40·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.680·5-s + 0.408·6-s − 1.28·7-s − 0.353·8-s + 0.333·9-s + 0.481·10-s + 0.872·11-s − 0.288·12-s + 0.277·13-s + 0.910·14-s + 0.393·15-s + 0.250·16-s + 0.880·17-s − 0.235·18-s + 1.91·19-s − 0.340·20-s + 0.743·21-s − 0.616·22-s − 1.77·23-s + 0.204·24-s − 0.536·25-s − 0.196·26-s − 0.192·27-s − 0.643·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$8034$$    =    $$2 \cdot 3 \cdot 13 \cdot 103$$ Sign: $1$ Motivic weight: $$1$$ Character: $\chi_{8034} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 8034,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.7475195233$$ $$L(\frac12)$$ $$\approx$$ $$0.7475195233$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + T$$
13 $$1 - T$$
103 $$1 + T$$
good5 $$1 + 1.52T + 5T^{2}$$
7 $$1 + 3.40T + 7T^{2}$$
11 $$1 - 2.89T + 11T^{2}$$
17 $$1 - 3.63T + 17T^{2}$$
19 $$1 - 8.33T + 19T^{2}$$
23 $$1 + 8.49T + 23T^{2}$$
29 $$1 + 1.96T + 29T^{2}$$
31 $$1 - 2.49T + 31T^{2}$$
37 $$1 - 5.76T + 37T^{2}$$
41 $$1 - 1.30T + 41T^{2}$$
43 $$1 + 6.49T + 43T^{2}$$
47 $$1 - 4.75T + 47T^{2}$$
53 $$1 - 7.93T + 53T^{2}$$
59 $$1 + 7.98T + 59T^{2}$$
61 $$1 - 8.78T + 61T^{2}$$
67 $$1 - 3.78T + 67T^{2}$$
71 $$1 - 12.6T + 71T^{2}$$
73 $$1 + 8.45T + 73T^{2}$$
79 $$1 - 3.93T + 79T^{2}$$
83 $$1 - 11.4T + 83T^{2}$$
89 $$1 + 15.2T + 89T^{2}$$
97 $$1 + 14.2T + 97T^{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

−7.74255313003321294245922098627, −7.24677405307463966901276924855, −6.45281284390416049510489348230, −5.96818519169025333390077407125, −5.27079222580976633789216686351, −3.91287143156017259403074175054, −3.68158742548686267866313379530, −2.70032855286170308791352631450, −1.39911162262814804442677282062, −0.52975759548789676351445315703, 0.52975759548789676351445315703, 1.39911162262814804442677282062, 2.70032855286170308791352631450, 3.68158742548686267866313379530, 3.91287143156017259403074175054, 5.27079222580976633789216686351, 5.96818519169025333390077407125, 6.45281284390416049510489348230, 7.24677405307463966901276924855, 7.74255313003321294245922098627