# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + 3-s + 4-s − 2.57·5-s − 6-s − 4.81·7-s − 8-s + 9-s + 2.57·10-s + 1.68·11-s + 12-s − 13-s + 4.81·14-s − 2.57·15-s + 16-s + 6.18·17-s − 18-s + 0.771·19-s − 2.57·20-s − 4.81·21-s − 1.68·22-s − 2.46·23-s − 24-s + 1.61·25-s + 26-s + 27-s − 4.81·28-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.15·5-s − 0.408·6-s − 1.81·7-s − 0.353·8-s + 0.333·9-s + 0.813·10-s + 0.506·11-s + 0.288·12-s − 0.277·13-s + 1.28·14-s − 0.664·15-s + 0.250·16-s + 1.49·17-s − 0.235·18-s + 0.176·19-s − 0.575·20-s − 1.05·21-s − 0.358·22-s − 0.513·23-s − 0.204·24-s + 0.323·25-s + 0.196·26-s + 0.192·27-s − 0.909·28-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: $$1$$ Selberg data: $$(2,\ 8034,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 + 2.57T + 5T^{2}$$
7 $$1 + 4.81T + 7T^{2}$$
11 $$1 - 1.68T + 11T^{2}$$
17 $$1 - 6.18T + 17T^{2}$$
19 $$1 - 0.771T + 19T^{2}$$
23 $$1 + 2.46T + 23T^{2}$$
29 $$1 + 4.09T + 29T^{2}$$
31 $$1 + 7.36T + 31T^{2}$$
37 $$1 - 7.10T + 37T^{2}$$
41 $$1 - 11.9T + 41T^{2}$$
43 $$1 - 4.87T + 43T^{2}$$
47 $$1 + 7.45T + 47T^{2}$$
53 $$1 - 4.54T + 53T^{2}$$
59 $$1 + 5.91T + 59T^{2}$$
61 $$1 + 0.0881T + 61T^{2}$$
67 $$1 + 9.81T + 67T^{2}$$
71 $$1 + 6.15T + 71T^{2}$$
73 $$1 - 16.5T + 73T^{2}$$
79 $$1 + 3.92T + 79T^{2}$$
83 $$1 + 1.22T + 83T^{2}$$
89 $$1 - 6.45T + 89T^{2}$$
97 $$1 + 8.09T + 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

−7.51518700566527926179778611806, −7.17512795077838866761794970973, −6.18790844689412588354932227330, −5.72008324008686738316394211041, −4.32143172631233108310964227148, −3.58553753459048357442804218170, −3.26482777101139502590695638719, −2.32986902366311676861853963242, −0.989210729909969253617787556756, 0, 0.989210729909969253617787556756, 2.32986902366311676861853963242, 3.26482777101139502590695638719, 3.58553753459048357442804218170, 4.32143172631233108310964227148, 5.72008324008686738316394211041, 6.18790844689412588354932227330, 7.17512795077838866761794970973, 7.51518700566527926179778611806