# Properties

 Label 2-76-19.11-c1-0-0 Degree $2$ Conductor $76$ Sign $0.910 + 0.412i$ Analytic cond. $0.606863$ Root an. cond. $0.779014$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (1 + 1.73i)9-s − 4·11-s + (0.5 + 0.866i)13-s + (−0.499 − 0.866i)15-s + (−1.5 + 2.59i)17-s + (−4 + 1.73i)19-s + (−2.5 − 4.33i)23-s + (2 + 3.46i)25-s + 5·27-s + (−3.5 − 6.06i)29-s + 4·31-s + (−2 + 3.46i)33-s + 10·37-s + ⋯
 L(s)  = 1 + (0.288 − 0.499i)3-s + (0.223 − 0.387i)5-s + (0.333 + 0.577i)9-s − 1.20·11-s + (0.138 + 0.240i)13-s + (−0.129 − 0.223i)15-s + (−0.363 + 0.630i)17-s + (−0.917 + 0.397i)19-s + (−0.521 − 0.902i)23-s + (0.400 + 0.692i)25-s + 0.962·27-s + (−0.649 − 1.12i)29-s + 0.718·31-s + (−0.348 + 0.603i)33-s + 1.64·37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$76$$    =    $$2^{2} \cdot 19$$ Sign: $0.910 + 0.412i$ Analytic conductor: $$0.606863$$ Root analytic conductor: $$0.779014$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{76} (49, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 76,\ (\ :1/2),\ 0.910 + 0.412i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.986847 - 0.213284i$$ $$L(\frac12)$$ $$\approx$$ $$0.986847 - 0.213284i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
19 $$1 + (4 - 1.73i)T$$
good3 $$1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2}$$
5 $$1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2}$$
7 $$1 + 7T^{2}$$
11 $$1 + 4T + 11T^{2}$$
13 $$1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2}$$
23 $$1 + (2.5 + 4.33i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (3.5 + 6.06i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 - 4T + 31T^{2}$$
37 $$1 - 10T + 37T^{2}$$
41 $$1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (-2.5 + 4.33i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (-3.5 - 6.06i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (5.5 + 9.52i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + (5.5 - 9.52i)T + (-35.5 - 61.4i)T^{2}$$
73 $$1 + (7.5 - 12.9i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)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

−14.29974575719456045326622773158, −13.11069918964448992657991348226, −12.74655459434758243945139301072, −11.06774901692491150864147072101, −10.03822225854232995061329335650, −8.523628327780524062372942515050, −7.65381165678650911206172693551, −6.09883239151249176774240439136, −4.53039496134745697616897108822, −2.23275176588073520095478123304, 2.88959967239244464973068077936, 4.60432761726607850681992359706, 6.24816901613009100736364334993, 7.68782966426915268740231684369, 9.083872573304780836458674175901, 10.13794998726485266646247897559, 11.07130322013113521553593114138, 12.56425310513379079331226375170, 13.54349859961333264765282700419, 14.76586366114238299366274732007