# Properties

 Label 2-966-161.10-c1-0-10 Degree $2$ Conductor $966$ Sign $0.353 - 0.935i$ Analytic cond. $7.71354$ Root an. cond. $2.77732$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.327 − 0.945i)2-s + (0.458 + 0.888i)3-s + (−0.786 − 0.618i)4-s + (−0.000133 + 1.27e−5i)5-s + (0.989 − 0.142i)6-s + (1.26 + 2.32i)7-s + (−0.841 + 0.540i)8-s + (−0.580 + 0.814i)9-s + (−3.16e−5 + 0.000130i)10-s + (−2.18 + 0.757i)11-s + (0.189 − 0.981i)12-s + (1.48 + 5.06i)13-s + (2.61 − 0.430i)14-s + (−7.26e−5 − 0.000113i)15-s + (0.235 + 0.971i)16-s + (−4.72 − 1.89i)17-s + ⋯
 L(s)  = 1 + (0.231 − 0.668i)2-s + (0.264 + 0.513i)3-s + (−0.393 − 0.309i)4-s + (−5.98e−5 + 5.71e−6i)5-s + (0.404 − 0.0580i)6-s + (0.476 + 0.879i)7-s + (−0.297 + 0.191i)8-s + (−0.193 + 0.271i)9-s + (−1.00e−5 + 4.12e−5i)10-s + (−0.659 + 0.228i)11-s + (0.0546 − 0.283i)12-s + (0.412 + 1.40i)13-s + (0.697 − 0.114i)14-s + (−1.87e−5 − 2.91e−5i)15-s + (0.0589 + 0.242i)16-s + (−1.14 − 0.458i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$966$$    =    $$2 \cdot 3 \cdot 7 \cdot 23$$ Sign: $0.353 - 0.935i$ Analytic conductor: $$7.71354$$ Root analytic conductor: $$2.77732$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{966} (493, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 966,\ (\ :1/2),\ 0.353 - 0.935i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.23228 + 0.851538i$$ $$L(\frac12)$$ $$\approx$$ $$1.23228 + 0.851538i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.327 + 0.945i)T$$
3 $$1 + (-0.458 - 0.888i)T$$
7 $$1 + (-1.26 - 2.32i)T$$
23 $$1 + (-3.59 - 3.17i)T$$
good5 $$1 + (0.000133 - 1.27e-5i)T + (4.90 - 0.946i)T^{2}$$
11 $$1 + (2.18 - 0.757i)T + (8.64 - 6.79i)T^{2}$$
13 $$1 + (-1.48 - 5.06i)T + (-10.9 + 7.02i)T^{2}$$
17 $$1 + (4.72 + 1.89i)T + (12.3 + 11.7i)T^{2}$$
19 $$1 + (2.45 - 0.981i)T + (13.7 - 13.1i)T^{2}$$
29 $$1 + (0.0478 + 0.332i)T + (-27.8 + 8.17i)T^{2}$$
31 $$1 + (-0.479 + 0.0228i)T + (30.8 - 2.94i)T^{2}$$
37 $$1 + (-7.64 - 5.44i)T + (12.1 + 34.9i)T^{2}$$
41 $$1 + (5.37 + 2.45i)T + (26.8 + 30.9i)T^{2}$$
43 $$1 + (-1.37 + 2.14i)T + (-17.8 - 39.1i)T^{2}$$
47 $$1 + (-1.41 + 0.817i)T + (23.5 - 40.7i)T^{2}$$
53 $$1 + (-7.60 - 7.97i)T + (-2.52 + 52.9i)T^{2}$$
59 $$1 + (0.946 + 0.229i)T + (52.4 + 27.0i)T^{2}$$
61 $$1 + (-5.85 - 3.01i)T + (35.3 + 49.6i)T^{2}$$
67 $$1 + (-0.312 - 1.62i)T + (-62.2 + 24.9i)T^{2}$$
71 $$1 + (-2.20 - 2.54i)T + (-10.1 + 70.2i)T^{2}$$
73 $$1 + (-8.53 + 10.8i)T + (-17.2 - 70.9i)T^{2}$$
79 $$1 + (-1.82 + 1.90i)T + (-3.75 - 78.9i)T^{2}$$
83 $$1 + (2.17 + 4.75i)T + (-54.3 + 62.7i)T^{2}$$
89 $$1 + (0.354 - 7.43i)T + (-88.5 - 8.45i)T^{2}$$
97 $$1 + (7.72 - 16.9i)T + (-63.5 - 73.3i)T^{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

−10.20005614045243921523853866821, −9.262701361255510218846968868480, −8.871795200637956005481096747344, −7.909834612471419749976480092387, −6.67163144368746580113856188491, −5.59987376765114131060944016501, −4.72121072130280153922164416357, −3.98215523124711468455405989963, −2.65248536768127760166462667133, −1.86125002293713048936866202005, 0.60960658181908175762143311501, 2.37563692138627286770660352262, 3.63626353582881997852220951337, 4.61584908967294455396204888985, 5.63949366434759152310780688645, 6.53339332081747900768592466921, 7.35697517102502937576929113855, 8.189904031409061150353001319502, 8.527858897796249121780938411663, 9.844826822037738501795296542370