# Properties

 Label 2-980-140.139-c1-0-96 Degree $2$ Conductor $980$ Sign $-0.206 + 0.978i$ Analytic cond. $7.82533$ Root an. cond. $2.79738$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.14 − 0.826i)2-s + 1.52i·3-s + (0.632 − 1.89i)4-s + (0.0967 − 2.23i)5-s + (1.25 + 1.74i)6-s + (−0.843 − 2.69i)8-s + 0.679·9-s + (−1.73 − 2.64i)10-s − 4.56i·11-s + (2.89 + 0.963i)12-s − 2.19·13-s + (3.40 + 0.147i)15-s + (−3.19 − 2.40i)16-s − 6.22·17-s + (0.779 − 0.561i)18-s + 3.83·19-s + ⋯
 L(s)  = 1 + (0.811 − 0.584i)2-s + 0.879i·3-s + (0.316 − 0.948i)4-s + (0.0432 − 0.999i)5-s + (0.514 + 0.713i)6-s + (−0.298 − 0.954i)8-s + 0.226·9-s + (−0.549 − 0.835i)10-s − 1.37i·11-s + (0.834 + 0.278i)12-s − 0.608·13-s + (0.878 + 0.0380i)15-s + (−0.799 − 0.600i)16-s − 1.50·17-s + (0.183 − 0.132i)18-s + 0.879·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$980$$    =    $$2^{2} \cdot 5 \cdot 7^{2}$$ Sign: $-0.206 + 0.978i$ Analytic conductor: $$7.82533$$ Root analytic conductor: $$2.79738$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{980} (979, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 980,\ (\ :1/2),\ -0.206 + 0.978i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.47069 - 1.81329i$$ $$L(\frac12)$$ $$\approx$$ $$1.47069 - 1.81329i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1.14 + 0.826i)T$$
5 $$1 + (-0.0967 + 2.23i)T$$
7 $$1$$
good3 $$1 - 1.52iT - 3T^{2}$$
11 $$1 + 4.56iT - 11T^{2}$$
13 $$1 + 2.19T + 13T^{2}$$
17 $$1 + 6.22T + 17T^{2}$$
19 $$1 - 3.83T + 19T^{2}$$
23 $$1 - 0.430T + 23T^{2}$$
29 $$1 + 0.473T + 29T^{2}$$
31 $$1 - 7.59T + 31T^{2}$$
37 $$1 + 8.44iT - 37T^{2}$$
41 $$1 + 1.45iT - 41T^{2}$$
43 $$1 - 8.58T + 43T^{2}$$
47 $$1 + 4.48iT - 47T^{2}$$
53 $$1 - 9.23iT - 53T^{2}$$
59 $$1 + 3.13T + 59T^{2}$$
61 $$1 + 5.71iT - 61T^{2}$$
67 $$1 - 14.9T + 67T^{2}$$
71 $$1 - 4.57iT - 71T^{2}$$
73 $$1 - 12.2T + 73T^{2}$$
79 $$1 - 6.20iT - 79T^{2}$$
83 $$1 - 7.69iT - 83T^{2}$$
89 $$1 - 9.32iT - 89T^{2}$$
97 $$1 + 9.05T + 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

−9.707014206296338958331955654332, −9.287328764027595688252145073942, −8.376659648928959368317311451859, −7.04913345054060719718509125250, −5.90258789872149738143769009450, −5.15304697045377402762981261006, −4.40063816118343567452818661534, −3.67219821074116647303507570455, −2.40527950664686895965153624787, −0.825828449723787310694362672260, 2.02134258650452765597460491954, 2.78980971191265311540899910876, 4.18884539763060867832554047917, 4.96372902290262285551229615499, 6.36071163432435031720456734047, 6.77370063842249558271217520304, 7.42540235601784689150584550247, 8.050881579850480061489194841235, 9.422083344440308807724647626924, 10.25044911313293856488680672593