Properties

Label 2-177-1.1-c3-0-5
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.21·2-s − 3·3-s + 9.78·4-s + 17.5·5-s + 12.6·6-s + 14.8·7-s − 7.53·8-s + 9·9-s − 74.0·10-s − 18.9·11-s − 29.3·12-s + 31.7·13-s − 62.8·14-s − 52.6·15-s − 46.5·16-s − 84.1·17-s − 37.9·18-s + 126.·19-s + 171.·20-s − 44.6·21-s + 79.7·22-s − 21.1·23-s + 22.6·24-s + 183.·25-s − 133.·26-s − 27·27-s + 145.·28-s + ⋯
L(s)  = 1  − 1.49·2-s − 0.577·3-s + 1.22·4-s + 1.57·5-s + 0.860·6-s + 0.804·7-s − 0.333·8-s + 0.333·9-s − 2.34·10-s − 0.518·11-s − 0.706·12-s + 0.677·13-s − 1.19·14-s − 0.906·15-s − 0.726·16-s − 1.20·17-s − 0.497·18-s + 1.52·19-s + 1.92·20-s − 0.464·21-s + 0.772·22-s − 0.191·23-s + 0.192·24-s + 1.46·25-s − 1.01·26-s − 0.192·27-s + 0.984·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9744982040\)
\(L(\frac12)\) \(\approx\) \(0.9744982040\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
59 \( 1 + 59T \)
good2 \( 1 + 4.21T + 8T^{2} \)
5 \( 1 - 17.5T + 125T^{2} \)
7 \( 1 - 14.8T + 343T^{2} \)
11 \( 1 + 18.9T + 1.33e3T^{2} \)
13 \( 1 - 31.7T + 2.19e3T^{2} \)
17 \( 1 + 84.1T + 4.91e3T^{2} \)
19 \( 1 - 126.T + 6.85e3T^{2} \)
23 \( 1 + 21.1T + 1.21e4T^{2} \)
29 \( 1 + 12.6T + 2.43e4T^{2} \)
31 \( 1 - 215.T + 2.97e4T^{2} \)
37 \( 1 + 3.62T + 5.06e4T^{2} \)
41 \( 1 + 381.T + 6.89e4T^{2} \)
43 \( 1 - 168.T + 7.95e4T^{2} \)
47 \( 1 - 613.T + 1.03e5T^{2} \)
53 \( 1 - 270.T + 1.48e5T^{2} \)
61 \( 1 - 311.T + 2.26e5T^{2} \)
67 \( 1 - 375.T + 3.00e5T^{2} \)
71 \( 1 + 987.T + 3.57e5T^{2} \)
73 \( 1 + 359.T + 3.89e5T^{2} \)
79 \( 1 - 933.T + 4.93e5T^{2} \)
83 \( 1 - 1.26e3T + 5.71e5T^{2} \)
89 \( 1 - 273.T + 7.04e5T^{2} \)
97 \( 1 + 897.T + 9.12e5T^{2} \)
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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

−11.78605938195553725856045506059, −10.84133969463592363413002744034, −10.17585446946839226603787604101, −9.303942910179749152973314130608, −8.392637150619768251435758109005, −7.16462251050260520026012806076, −6.03855946941561769933655651333, −4.91729102188014468784388875909, −2.20894023297450445440531222765, −1.07232283918971311365301094771, 1.07232283918971311365301094771, 2.20894023297450445440531222765, 4.91729102188014468784388875909, 6.03855946941561769933655651333, 7.16462251050260520026012806076, 8.392637150619768251435758109005, 9.303942910179749152973314130608, 10.17585446946839226603787604101, 10.84133969463592363413002744034, 11.78605938195553725856045506059

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