Properties

Label 2-177-1.1-c5-0-17
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.96·2-s + 9·3-s + 3.52·4-s + 65.6·5-s − 53.6·6-s + 119.·7-s + 169.·8-s + 81·9-s − 391.·10-s + 237.·11-s + 31.7·12-s + 999.·13-s − 709.·14-s + 590.·15-s − 1.12e3·16-s − 1.27e3·17-s − 482.·18-s + 1.42e3·19-s + 231.·20-s + 1.07e3·21-s − 1.41e3·22-s + 546.·23-s + 1.52e3·24-s + 1.18e3·25-s − 5.95e3·26-s + 729·27-s + 419.·28-s + ⋯
L(s)  = 1  − 1.05·2-s + 0.577·3-s + 0.110·4-s + 1.17·5-s − 0.608·6-s + 0.918·7-s + 0.937·8-s + 0.333·9-s − 1.23·10-s + 0.592·11-s + 0.0635·12-s + 1.63·13-s − 0.967·14-s + 0.677·15-s − 1.09·16-s − 1.07·17-s − 0.351·18-s + 0.906·19-s + 0.129·20-s + 0.530·21-s − 0.623·22-s + 0.215·23-s + 0.541·24-s + 0.378·25-s − 1.72·26-s + 0.192·27-s + 0.101·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.023806391\)
\(L(\frac12)\) \(\approx\) \(2.023806391\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9T \)
59 \( 1 + 3.48e3T \)
good2 \( 1 + 5.96T + 32T^{2} \)
5 \( 1 - 65.6T + 3.12e3T^{2} \)
7 \( 1 - 119.T + 1.68e4T^{2} \)
11 \( 1 - 237.T + 1.61e5T^{2} \)
13 \( 1 - 999.T + 3.71e5T^{2} \)
17 \( 1 + 1.27e3T + 1.41e6T^{2} \)
19 \( 1 - 1.42e3T + 2.47e6T^{2} \)
23 \( 1 - 546.T + 6.43e6T^{2} \)
29 \( 1 + 1.54e3T + 2.05e7T^{2} \)
31 \( 1 + 7.56e3T + 2.86e7T^{2} \)
37 \( 1 + 1.01e3T + 6.93e7T^{2} \)
41 \( 1 - 1.32e4T + 1.15e8T^{2} \)
43 \( 1 + 2.17e4T + 1.47e8T^{2} \)
47 \( 1 - 2.30e4T + 2.29e8T^{2} \)
53 \( 1 - 1.90e4T + 4.18e8T^{2} \)
61 \( 1 + 7.23e3T + 8.44e8T^{2} \)
67 \( 1 - 6.98e4T + 1.35e9T^{2} \)
71 \( 1 - 1.78e4T + 1.80e9T^{2} \)
73 \( 1 - 9.08e3T + 2.07e9T^{2} \)
79 \( 1 - 3.14e4T + 3.07e9T^{2} \)
83 \( 1 + 1.02e5T + 3.93e9T^{2} \)
89 \( 1 - 1.04e5T + 5.58e9T^{2} \)
97 \( 1 + 1.17e5T + 8.58e9T^{2} \)
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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

−11.34194255764223341647508751966, −10.62476041256158269869529120292, −9.430196724757437899891888399993, −8.944130306332400848352492250360, −8.062018692526054879521726836954, −6.81715657767036138761339413074, −5.42727246619447774829103829280, −3.96616703495734610580077691878, −1.97859293909277273559541661128, −1.17858681667078262042467960945, 1.17858681667078262042467960945, 1.97859293909277273559541661128, 3.96616703495734610580077691878, 5.42727246619447774829103829280, 6.81715657767036138761339413074, 8.062018692526054879521726836954, 8.944130306332400848352492250360, 9.430196724757437899891888399993, 10.62476041256158269869529120292, 11.34194255764223341647508751966

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