Properties

Degree $2$
Conductor $177$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.86·2-s − 3-s + 1.46·4-s − 3.32·5-s + 1.86·6-s + 1.13·7-s + 8-s + 9-s + 6.18·10-s + 0.398·11-s − 1.46·12-s + 2.39·13-s − 2.11·14-s + 3.32·15-s − 4.78·16-s + 7.10·17-s − 1.86·18-s + 1.53·19-s − 4.86·20-s − 1.13·21-s − 0.740·22-s + 3.25·23-s − 24-s + 6.04·25-s − 4.46·26-s − 27-s + 1.66·28-s + ⋯
L(s)  = 1  − 1.31·2-s − 0.577·3-s + 0.731·4-s − 1.48·5-s + 0.759·6-s + 0.430·7-s + 0.353·8-s + 0.333·9-s + 1.95·10-s + 0.120·11-s − 0.422·12-s + 0.665·13-s − 0.566·14-s + 0.858·15-s − 1.19·16-s + 1.72·17-s − 0.438·18-s + 0.352·19-s − 1.08·20-s − 0.248·21-s − 0.157·22-s + 0.679·23-s − 0.204·24-s + 1.20·25-s − 0.875·26-s − 0.192·27-s + 0.314·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.415659\)
\(L(\frac12)\) \(\approx\) \(0.415659\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
59 \( 1 - T \)
good2 \( 1 + 1.86T + 2T^{2} \)
5 \( 1 + 3.32T + 5T^{2} \)
7 \( 1 - 1.13T + 7T^{2} \)
11 \( 1 - 0.398T + 11T^{2} \)
13 \( 1 - 2.39T + 13T^{2} \)
17 \( 1 - 7.10T + 17T^{2} \)
19 \( 1 - 1.53T + 19T^{2} \)
23 \( 1 - 3.25T + 23T^{2} \)
29 \( 1 + 3.93T + 29T^{2} \)
31 \( 1 - 7.78T + 31T^{2} \)
37 \( 1 - 1.25T + 37T^{2} \)
41 \( 1 + 7.50T + 41T^{2} \)
43 \( 1 + 9.69T + 43T^{2} \)
47 \( 1 - 8.71T + 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
61 \( 1 - 0.989T + 61T^{2} \)
67 \( 1 - 5.45T + 67T^{2} \)
71 \( 1 - 15.0T + 71T^{2} \)
73 \( 1 - 4.73T + 73T^{2} \)
79 \( 1 - 7.04T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−12.15115091323008454373695491675, −11.52832314049207541659394202048, −10.71237798456219590306607710559, −9.721217379621078642055548092576, −8.401406427558736789534259397578, −7.85009310363940692148093872438, −6.87134282413878860829818389583, −5.10041782871437357986933036727, −3.70709883862564397941779945184, −0.983967072382549312772776173028, 0.983967072382549312772776173028, 3.70709883862564397941779945184, 5.10041782871437357986933036727, 6.87134282413878860829818389583, 7.85009310363940692148093872438, 8.401406427558736789534259397578, 9.721217379621078642055548092576, 10.71237798456219590306607710559, 11.52832314049207541659394202048, 12.15115091323008454373695491675

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