# Properties

 Label 2-5070-13.12-c1-0-12 Degree $2$ Conductor $5070$ Sign $0.999 - 0.0304i$ Analytic cond. $40.4841$ Root an. cond. $6.36271$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s − 3-s − 4-s − i·5-s + i·6-s − 1.69i·7-s + i·8-s + 9-s − 10-s − 4.55i·11-s + 12-s − 1.69·14-s + i·15-s + 16-s − 2.35·17-s − i·18-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s − 0.639i·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s − 1.37i·11-s + 0.288·12-s − 0.452·14-s + 0.258i·15-s + 0.250·16-s − 0.571·17-s − 0.235i·18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$5070$$    =    $$2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Sign: $0.999 - 0.0304i$ Analytic conductor: $$40.4841$$ Root analytic conductor: $$6.36271$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{5070} (1351, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 5070,\ (\ :1/2),\ 0.999 - 0.0304i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.8160912384$$ $$L(\frac12)$$ $$\approx$$ $$0.8160912384$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
3 $$1 + T$$
5 $$1 + iT$$
13 $$1$$
good7 $$1 + 1.69iT - 7T^{2}$$
11 $$1 + 4.55iT - 11T^{2}$$
17 $$1 + 2.35T + 17T^{2}$$
19 $$1 - 6.51iT - 19T^{2}$$
23 $$1 + 8.94T + 23T^{2}$$
29 $$1 - 9.07T + 29T^{2}$$
31 $$1 - 10.6iT - 31T^{2}$$
37 $$1 - 6.18iT - 37T^{2}$$
41 $$1 + 3.00iT - 41T^{2}$$
43 $$1 - 4.93T + 43T^{2}$$
47 $$1 + 4.28iT - 47T^{2}$$
53 $$1 + 3.40T + 53T^{2}$$
59 $$1 - 4.32iT - 59T^{2}$$
61 $$1 + 14.3T + 61T^{2}$$
67 $$1 - 3.24iT - 67T^{2}$$
71 $$1 - 14.9iT - 71T^{2}$$
73 $$1 + 6.72iT - 73T^{2}$$
79 $$1 - 5.67T + 79T^{2}$$
83 $$1 - 7.71iT - 83T^{2}$$
89 $$1 + 9.12iT - 89T^{2}$$
97 $$1 - 4.40iT - 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

−8.373155921936392215865095126674, −7.71067149517681271293787408442, −6.59321111774236030356562224266, −6.00581419016006499277032572703, −5.28252712175064401627976964208, −4.37681843133250048585484199756, −3.81180721996032855076429499071, −2.93487918029643805934258069774, −1.64974660127367491576258836081, −0.865356396450850205167519087393, 0.30340717774835447534491289212, 1.98485520790170391133114084274, 2.70181368618721728365467700752, 4.18217783411873223872517751340, 4.53837500518646834613308368130, 5.41753645083922910864165038063, 6.32016742724110061705281405754, 6.52774595925722188672868640611, 7.57282565500440446796273367690, 7.87422096569913188648727440239