Properties

Label 2-177-177.176-c5-0-18
Degree $2$
Conductor $177$
Sign $-0.650 - 0.759i$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11.8 − 10.1i)3-s − 32·4-s + 111. i·5-s + 113.·7-s + (37.4 − 240. i)9-s + (−378. + 324. i)12-s + (1.13e3 + 1.32e3i)15-s + 1.02e3·16-s + 683. i·17-s − 3.06e3·19-s − 3.57e3i·20-s + (1.34e3 − 1.14e3i)21-s − 9.32e3·25-s + (−1.99e3 − 3.22e3i)27-s − 3.62e3·28-s + 8.94e3i·29-s + ⋯
L(s)  = 1  + (0.759 − 0.650i)3-s − 4-s + 1.99i·5-s + 0.873·7-s + (0.153 − 0.988i)9-s + (−0.759 + 0.650i)12-s + (1.29 + 1.51i)15-s + 16-s + 0.573i·17-s − 1.94·19-s − 1.99i·20-s + (0.663 − 0.567i)21-s − 2.98·25-s + (−0.525 − 0.850i)27-s − 0.873·28-s + 1.97i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.650 - 0.759i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.650 - 0.759i$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ -0.650 - 0.759i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.235739929\)
\(L(\frac12)\) \(\approx\) \(1.235739929\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-11.8 + 10.1i)T \)
59 \( 1 - 2.67e4iT \)
good2 \( 1 + 32T^{2} \)
5 \( 1 - 111. iT - 3.12e3T^{2} \)
7 \( 1 - 113.T + 1.68e4T^{2} \)
11 \( 1 + 1.61e5T^{2} \)
13 \( 1 - 3.71e5T^{2} \)
17 \( 1 - 683. iT - 1.41e6T^{2} \)
19 \( 1 + 3.06e3T + 2.47e6T^{2} \)
23 \( 1 + 6.43e6T^{2} \)
29 \( 1 - 8.94e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.86e7T^{2} \)
37 \( 1 - 6.93e7T^{2} \)
41 \( 1 + 524. iT - 1.15e8T^{2} \)
43 \( 1 - 1.47e8T^{2} \)
47 \( 1 + 2.29e8T^{2} \)
53 \( 1 - 3.04e4iT - 4.18e8T^{2} \)
61 \( 1 - 8.44e8T^{2} \)
67 \( 1 - 1.35e9T^{2} \)
71 \( 1 + 5.94e4iT - 1.80e9T^{2} \)
73 \( 1 - 2.07e9T^{2} \)
79 \( 1 + 9.52e4T + 3.07e9T^{2} \)
83 \( 1 + 3.93e9T^{2} \)
89 \( 1 + 5.58e9T^{2} \)
97 \( 1 - 8.58e9T^{2} \)
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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

−12.31602112473714578560486821695, −10.95117470614282094040707181641, −10.29495426393076973185579000017, −8.931000952357777758696073334470, −8.049588177425549433208313755880, −7.09452088091934992560493726445, −6.06413557138978164724247350829, −4.19522742716935936313883447974, −3.09290071687403850225322026677, −1.79881452584896459338560862693, 0.35775265686656443075088849696, 1.87304051097798361934430712277, 4.15745704638152719725346799770, 4.57314034280603070126093001327, 5.50996238274063698729212169954, 8.124401706294861240995090120632, 8.349473905029746918382058117713, 9.246747296712957424834576522190, 10.01322406985471398090600689297, 11.52154894102574582906262426242

Graph of the $Z$-function along the critical line