# Properties

 Label 2-1260-28.27-c1-0-20 Degree $2$ Conductor $1260$ Sign $0.0661 - 0.997i$ Analytic cond. $10.0611$ Root an. cond. $3.17193$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.10 + 0.887i)2-s + (0.426 − 1.95i)4-s + i·5-s + (−0.391 − 2.61i)7-s + (1.26 + 2.53i)8-s + (−0.887 − 1.10i)10-s − 0.770i·11-s + 5.60i·13-s + (2.75 + 2.53i)14-s + (−3.63 − 1.66i)16-s − 0.503i·17-s − 1.63·19-s + (1.95 + 0.426i)20-s + (0.683 + 0.848i)22-s + 1.42i·23-s + ⋯
 L(s)  = 1 + (−0.778 + 0.627i)2-s + (0.213 − 0.977i)4-s + 0.447i·5-s + (−0.148 − 0.988i)7-s + (0.446 + 0.894i)8-s + (−0.280 − 0.348i)10-s − 0.232i·11-s + 1.55i·13-s + (0.735 + 0.677i)14-s + (−0.909 − 0.416i)16-s − 0.122i·17-s − 0.375·19-s + (0.436 + 0.0953i)20-s + (0.145 + 0.180i)22-s + 0.297i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0661 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0661 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1260$$    =    $$2^{2} \cdot 3^{2} \cdot 5 \cdot 7$$ Sign: $0.0661 - 0.997i$ Analytic conductor: $$10.0611$$ Root analytic conductor: $$3.17193$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1260} (811, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1260,\ (\ :1/2),\ 0.0661 - 0.997i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.9438744544$$ $$L(\frac12)$$ $$\approx$$ $$0.9438744544$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.10 - 0.887i)T$$
3 $$1$$
5 $$1 - iT$$
7 $$1 + (0.391 + 2.61i)T$$
good11 $$1 + 0.770iT - 11T^{2}$$
13 $$1 - 5.60iT - 13T^{2}$$
17 $$1 + 0.503iT - 17T^{2}$$
19 $$1 + 1.63T + 19T^{2}$$
23 $$1 - 1.42iT - 23T^{2}$$
29 $$1 + 5.03T + 29T^{2}$$
31 $$1 - 8.23T + 31T^{2}$$
37 $$1 - 10.1T + 37T^{2}$$
41 $$1 - 5.07iT - 41T^{2}$$
43 $$1 - 9.06iT - 43T^{2}$$
47 $$1 - 4.64T + 47T^{2}$$
53 $$1 + 0.455T + 53T^{2}$$
59 $$1 - 10.4T + 59T^{2}$$
61 $$1 - 3.32iT - 61T^{2}$$
67 $$1 - 8.70iT - 67T^{2}$$
71 $$1 + 10.8iT - 71T^{2}$$
73 $$1 + 2.29iT - 73T^{2}$$
79 $$1 + 2.56iT - 79T^{2}$$
83 $$1 - 13.9T + 83T^{2}$$
89 $$1 - 2.98iT - 89T^{2}$$
97 $$1 - 15.5iT - 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

−9.700378228338708621040478063655, −9.205878413489750413617931228469, −8.096284428211796089087244178595, −7.47339554867651658786236145680, −6.59890758381027059269763465904, −6.19015377693956172482044010855, −4.79719082907147946928063126925, −3.96766548118265862453364967365, −2.45369962033876810413811874945, −1.11679643586909628987565259826, 0.60680449142134696130237027535, 2.12726098354186889570762968682, 2.97006425332859230896274863153, 4.11002938902333089774302156733, 5.29320472100578437785879804941, 6.15577232695206997068675719451, 7.31851194721488602388654761317, 8.214536405483179530741204867682, 8.635925046689846701742241414702, 9.565916474008605340910824164552