# Properties

 Label 2-177-177.176-c3-0-9 Degree $2$ Conductor $177$ Sign $-0.952 + 0.303i$ Analytic cond. $10.4433$ Root an. cond. $3.23161$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.57 + 4.95i)3-s − 8·4-s + 14.3i·5-s + 5.45·7-s + (−22.0 + 15.6i)9-s + (−12.6 − 39.6i)12-s + (−71.0 + 22.6i)15-s + 64·16-s − 61.4i·17-s − 159.·19-s − 114. i·20-s + (8.59 + 27.0i)21-s − 80.8·25-s + (−111. − 84.4i)27-s − 43.6·28-s − 3.27i·29-s + ⋯
 L(s)  = 1 + (0.303 + 0.952i)3-s − 4-s + 1.28i·5-s + 0.294·7-s + (−0.816 + 0.578i)9-s + (−0.303 − 0.952i)12-s + (−1.22 + 0.389i)15-s + 16-s − 0.876i·17-s − 1.92·19-s − 1.28i·20-s + (0.0893 + 0.280i)21-s − 0.646·25-s + (−0.798 − 0.602i)27-s − 0.294·28-s − 0.0209i·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.303i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$177$$    =    $$3 \cdot 59$$ Sign: $-0.952 + 0.303i$ Analytic conductor: $$10.4433$$ Root analytic conductor: $$3.23161$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{177} (176, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 177,\ (\ :3/2),\ -0.952 + 0.303i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.117867 - 0.758907i$$ $$L(\frac12)$$ $$\approx$$ $$0.117867 - 0.758907i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-1.57 - 4.95i)T$$
59 $$1 - 453. iT$$
good2 $$1 + 8T^{2}$$
5 $$1 - 14.3iT - 125T^{2}$$
7 $$1 - 5.45T + 343T^{2}$$
11 $$1 + 1.33e3T^{2}$$
13 $$1 - 2.19e3T^{2}$$
17 $$1 + 61.4iT - 4.91e3T^{2}$$
19 $$1 + 159.T + 6.85e3T^{2}$$
23 $$1 + 1.21e4T^{2}$$
29 $$1 + 3.27iT - 2.43e4T^{2}$$
31 $$1 - 2.97e4T^{2}$$
37 $$1 - 5.06e4T^{2}$$
41 $$1 - 450. iT - 6.89e4T^{2}$$
43 $$1 - 7.95e4T^{2}$$
47 $$1 + 1.03e5T^{2}$$
53 $$1 + 66.3iT - 1.48e5T^{2}$$
61 $$1 - 2.26e5T^{2}$$
67 $$1 - 3.00e5T^{2}$$
71 $$1 - 1.18e3iT - 3.57e5T^{2}$$
73 $$1 - 3.89e5T^{2}$$
79 $$1 - 1.39e3T + 4.93e5T^{2}$$
83 $$1 + 5.71e5T^{2}$$
89 $$1 + 7.04e5T^{2}$$
97 $$1 - 9.12e5T^{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

−12.99709166017941076698722672525, −11.47094680310679674245537816486, −10.61301895567652623012931867625, −9.868583732353880663384382388751, −8.850156423830596864415932079742, −7.88568581792547979547884420356, −6.38726489432904388152097064027, −4.96808870443704163107983524371, −3.94397026756928190544901243350, −2.70309303190026504760401059898, 0.34200826286994347660510959214, 1.76306720346853333465005933389, 3.93232725477115576020955822793, 5.08288369691708611030722406455, 6.29291062911492051506644527401, 7.941712549700444723198189741399, 8.567013926483412586395556568972, 9.221034150226689550101459524693, 10.69102150833420392225325246239, 12.23481228606290600691422484774