Properties

Label 2-177-1.1-c5-0-32
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.44·2-s + 9·3-s + 57.2·4-s + 13.7·5-s − 85.0·6-s + 67.4·7-s − 238.·8-s + 81·9-s − 130.·10-s − 138.·11-s + 514.·12-s − 485.·13-s − 637.·14-s + 124.·15-s + 417.·16-s − 2.22e3·17-s − 765.·18-s + 1.85e3·19-s + 788.·20-s + 607.·21-s + 1.31e3·22-s − 1.00e3·23-s − 2.14e3·24-s − 2.93e3·25-s + 4.58e3·26-s + 729·27-s + 3.86e3·28-s + ⋯
L(s)  = 1  − 1.66·2-s + 0.577·3-s + 1.78·4-s + 0.246·5-s − 0.963·6-s + 0.520·7-s − 1.31·8-s + 0.333·9-s − 0.411·10-s − 0.345·11-s + 1.03·12-s − 0.796·13-s − 0.869·14-s + 0.142·15-s + 0.407·16-s − 1.86·17-s − 0.556·18-s + 1.17·19-s + 0.440·20-s + 0.300·21-s + 0.577·22-s − 0.397·23-s − 0.759·24-s − 0.939·25-s + 1.32·26-s + 0.192·27-s + 0.930·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 9.44T + 32T^{2} \)
5 \( 1 - 13.7T + 3.12e3T^{2} \)
7 \( 1 - 67.4T + 1.68e4T^{2} \)
11 \( 1 + 138.T + 1.61e5T^{2} \)
13 \( 1 + 485.T + 3.71e5T^{2} \)
17 \( 1 + 2.22e3T + 1.41e6T^{2} \)
19 \( 1 - 1.85e3T + 2.47e6T^{2} \)
23 \( 1 + 1.00e3T + 6.43e6T^{2} \)
29 \( 1 - 1.53e3T + 2.05e7T^{2} \)
31 \( 1 - 5.92e3T + 2.86e7T^{2} \)
37 \( 1 - 1.07e3T + 6.93e7T^{2} \)
41 \( 1 + 1.05e4T + 1.15e8T^{2} \)
43 \( 1 - 1.50e4T + 1.47e8T^{2} \)
47 \( 1 + 1.49e4T + 2.29e8T^{2} \)
53 \( 1 + 4.08e4T + 4.18e8T^{2} \)
61 \( 1 + 3.42e4T + 8.44e8T^{2} \)
67 \( 1 - 1.84e4T + 1.35e9T^{2} \)
71 \( 1 - 1.76e4T + 1.80e9T^{2} \)
73 \( 1 + 2.56e4T + 2.07e9T^{2} \)
79 \( 1 + 1.35e4T + 3.07e9T^{2} \)
83 \( 1 + 8.03e4T + 3.93e9T^{2} \)
89 \( 1 - 4.37e4T + 5.58e9T^{2} \)
97 \( 1 - 6.76e4T + 8.58e9T^{2} \)
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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

−10.99869929413639579742633377223, −9.945792775628811512430488280861, −9.315626587138081095158341380804, −8.283026530162422210668191614809, −7.59836023838742052563712896760, −6.50477697751072135974423782672, −4.69618629153444525947846709132, −2.64416288601302314133943915513, −1.61857771753790420162371538831, 0, 1.61857771753790420162371538831, 2.64416288601302314133943915513, 4.69618629153444525947846709132, 6.50477697751072135974423782672, 7.59836023838742052563712896760, 8.283026530162422210668191614809, 9.315626587138081095158341380804, 9.945792775628811512430488280861, 10.99869929413639579742633377223

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