# Properties

 Label 2-76-4.3-c2-0-8 Degree $2$ Conductor $76$ Sign $0.436 - 0.899i$ Analytic cond. $2.07085$ Root an. cond. $1.43904$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.94 + 0.447i)2-s + 3.19i·3-s + (3.59 + 1.74i)4-s − 3.90·5-s + (−1.43 + 6.23i)6-s − 2.64i·7-s + (6.23 + 5.01i)8-s − 1.23·9-s + (−7.60 − 1.74i)10-s − 13.2i·11-s + (−5.58 + 11.5i)12-s + 1.60·13-s + (1.18 − 5.15i)14-s − 12.4i·15-s + (9.91 + 12.5i)16-s + 8.10·17-s + ⋯
 L(s)  = 1 + (0.974 + 0.223i)2-s + 1.06i·3-s + (0.899 + 0.436i)4-s − 0.780·5-s + (−0.238 + 1.03i)6-s − 0.378i·7-s + (0.779 + 0.626i)8-s − 0.137·9-s + (−0.760 − 0.174i)10-s − 1.20i·11-s + (−0.465 + 0.959i)12-s + 0.123·13-s + (0.0845 − 0.368i)14-s − 0.832i·15-s + (0.619 + 0.784i)16-s + 0.476·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$76$$    =    $$2^{2} \cdot 19$$ Sign: $0.436 - 0.899i$ Analytic conductor: $$2.07085$$ Root analytic conductor: $$1.43904$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{76} (39, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 76,\ (\ :1),\ 0.436 - 0.899i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.67127 + 1.04718i$$ $$L(\frac12)$$ $$\approx$$ $$1.67127 + 1.04718i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1.94 - 0.447i)T$$
19 $$1 - 4.35iT$$
good3 $$1 - 3.19iT - 9T^{2}$$
5 $$1 + 3.90T + 25T^{2}$$
7 $$1 + 2.64iT - 49T^{2}$$
11 $$1 + 13.2iT - 121T^{2}$$
13 $$1 - 1.60T + 169T^{2}$$
17 $$1 - 8.10T + 289T^{2}$$
23 $$1 + 38.2iT - 529T^{2}$$
29 $$1 + 51.1T + 841T^{2}$$
31 $$1 - 13.6iT - 961T^{2}$$
37 $$1 + 35.3T + 1.36e3T^{2}$$
41 $$1 - 8.06T + 1.68e3T^{2}$$
43 $$1 - 16.2iT - 1.84e3T^{2}$$
47 $$1 - 1.59iT - 2.20e3T^{2}$$
53 $$1 - 56.8T + 2.80e3T^{2}$$
59 $$1 - 106. iT - 3.48e3T^{2}$$
61 $$1 + 35.9T + 3.72e3T^{2}$$
67 $$1 - 47.5iT - 4.48e3T^{2}$$
71 $$1 + 125. iT - 5.04e3T^{2}$$
73 $$1 - 34.2T + 5.32e3T^{2}$$
79 $$1 - 71.1iT - 6.24e3T^{2}$$
83 $$1 + 149. iT - 6.88e3T^{2}$$
89 $$1 + 169.T + 7.92e3T^{2}$$
97 $$1 - 109.T + 9.40e3T^{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

−14.61817478156520369964272926220, −13.55144010006253019445719085047, −12.29578474942570283597827426532, −11.15758244361419508997785488936, −10.39840963082615081694052426746, −8.631660007393844863658705836710, −7.33525850884920181571913644323, −5.73038517966097620047823738092, −4.31427087735724631054194832323, −3.43291263748503298481147151087, 1.89647913782104993580287036651, 3.86245796768298382754820290981, 5.51882181003969479708021739931, 7.06172075844459048601310830793, 7.69718132623534560604355200338, 9.736925690919939146036487800368, 11.34532093716463591474445593196, 12.13445015173975691464552008637, 12.85783837680268847019466082082, 13.81487907523145300141753307341