# Properties

 Label 2-8470-1.1-c1-0-110 Degree $2$ Conductor $8470$ Sign $-1$ Analytic cond. $67.6332$ Root an. cond. $8.22394$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s − 2.90·3-s + 4-s − 5-s + 2.90·6-s + 7-s − 8-s + 5.43·9-s + 10-s − 2.90·12-s + 6.51·13-s − 14-s + 2.90·15-s + 16-s − 7.07·17-s − 5.43·18-s − 3.97·19-s − 20-s − 2.90·21-s + 9.39·23-s + 2.90·24-s + 25-s − 6.51·26-s − 7.08·27-s + 28-s − 1.17·29-s − 2.90·30-s + ⋯
 L(s)  = 1 − 0.707·2-s − 1.67·3-s + 0.5·4-s − 0.447·5-s + 1.18·6-s + 0.377·7-s − 0.353·8-s + 1.81·9-s + 0.316·10-s − 0.838·12-s + 1.80·13-s − 0.267·14-s + 0.750·15-s + 0.250·16-s − 1.71·17-s − 1.28·18-s − 0.911·19-s − 0.223·20-s − 0.633·21-s + 1.96·23-s + 0.592·24-s + 0.200·25-s − 1.27·26-s − 1.36·27-s + 0.188·28-s − 0.218·29-s − 0.530·30-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$8470$$    =    $$2 \cdot 5 \cdot 7 \cdot 11^{2}$$ Sign: $-1$ Analytic conductor: $$67.6332$$ Root analytic conductor: $$8.22394$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{8470} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 8470,\ (\ :1/2),\ -1)$$

## 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$$
5 $$1 + T$$
7 $$1 - T$$
11 $$1$$
good3 $$1 + 2.90T + 3T^{2}$$
13 $$1 - 6.51T + 13T^{2}$$
17 $$1 + 7.07T + 17T^{2}$$
19 $$1 + 3.97T + 19T^{2}$$
23 $$1 - 9.39T + 23T^{2}$$
29 $$1 + 1.17T + 29T^{2}$$
31 $$1 + 1.28T + 31T^{2}$$
37 $$1 + 1.00T + 37T^{2}$$
41 $$1 - 5.43T + 41T^{2}$$
43 $$1 - 11.2T + 43T^{2}$$
47 $$1 + 11.3T + 47T^{2}$$
53 $$1 + 7.75T + 53T^{2}$$
59 $$1 + 5.64T + 59T^{2}$$
61 $$1 + 6.33T + 61T^{2}$$
67 $$1 - 3.88T + 67T^{2}$$
71 $$1 + 0.259T + 71T^{2}$$
73 $$1 + 4.82T + 73T^{2}$$
79 $$1 + 14.3T + 79T^{2}$$
83 $$1 + 4.59T + 83T^{2}$$
89 $$1 + 1.43T + 89T^{2}$$
97 $$1 + 8.60T + 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

−7.26294481743855539195513727544, −6.66800164802151211181517815043, −6.22419486786882157462364961449, −5.55706464585278330694101018561, −4.60382254507348160385684868090, −4.20859357835914667679113697948, −3.04171055652687104135099270987, −1.73827978017805004058323021901, −0.978945424295542839997379250892, 0, 0.978945424295542839997379250892, 1.73827978017805004058323021901, 3.04171055652687104135099270987, 4.20859357835914667679113697948, 4.60382254507348160385684868090, 5.55706464585278330694101018561, 6.22419486786882157462364961449, 6.66800164802151211181517815043, 7.26294481743855539195513727544