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  − 0.254·2-s − 3-s − 1.93·4-s + 1.68·5-s + 0.254·6-s + 2.74·7-s + 8-s + 9-s − 0.427·10-s + 2.18·11-s + 1.93·12-s + 4.18·13-s − 0.697·14-s − 1.68·15-s + 3.61·16-s − 6.29·17-s − 0.254·18-s + 4.93·19-s − 3.25·20-s − 2.74·21-s − 0.556·22-s + 3.44·23-s − 24-s − 2.17·25-s − 1.06·26-s − 27-s − 5.31·28-s + ⋯
L(s)  = 1  − 0.179·2-s − 0.577·3-s − 0.967·4-s + 0.751·5-s + 0.103·6-s + 1.03·7-s + 0.353·8-s + 0.333·9-s − 0.135·10-s + 0.660·11-s + 0.558·12-s + 1.16·13-s − 0.186·14-s − 0.434·15-s + 0.904·16-s − 1.52·17-s − 0.0598·18-s + 1.13·19-s − 0.727·20-s − 0.599·21-s − 0.118·22-s + 0.718·23-s − 0.204·24-s − 0.434·25-s − 0.208·26-s − 0.192·27-s − 1.00·28-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

\(L(1)\) \(\approx\) \(0.950064\)
\(L(\frac12)\) \(\approx\) \(0.950064\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.254T + 2T^{2} \)
5 \( 1 - 1.68T + 5T^{2} \)
7 \( 1 - 2.74T + 7T^{2} \)
11 \( 1 - 2.18T + 11T^{2} \)
13 \( 1 - 4.18T + 13T^{2} \)
17 \( 1 + 6.29T + 17T^{2} \)
19 \( 1 - 4.93T + 19T^{2} \)
23 \( 1 - 3.44T + 23T^{2} \)
29 \( 1 + 9.12T + 29T^{2} \)
31 \( 1 + 0.616T + 31T^{2} \)
37 \( 1 - 1.44T + 37T^{2} \)
41 \( 1 - 4.10T + 41T^{2} \)
43 \( 1 - 8.53T + 43T^{2} \)
47 \( 1 + 6.48T + 47T^{2} \)
53 \( 1 + 0.664T + 53T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 + 9.93T + 67T^{2} \)
71 \( 1 - 6.82T + 71T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 + 1.17T + 79T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 - 1.97T + 89T^{2} \)
97 \( 1 - 3.33T + 97T^{2} \)
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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

−12.90021328052132812295433703083, −11.44093664029276226870754784110, −10.87566813003725267209946673304, −9.494066915536887870405735842648, −8.898106803057333287936314895230, −7.62603594227267634951765609978, −6.13976133051542895489123088615, −5.14592092532012328961583137997, −4.03097010048642152134062221981, −1.47181740843826448674411088050, 1.47181740843826448674411088050, 4.03097010048642152134062221981, 5.14592092532012328961583137997, 6.13976133051542895489123088615, 7.62603594227267634951765609978, 8.898106803057333287936314895230, 9.494066915536887870405735842648, 10.87566813003725267209946673304, 11.44093664029276226870754784110, 12.90021328052132812295433703083

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