Properties

Label 2-177-1.1-c13-0-114
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $189.798$
Root an. cond. $13.7767$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 156.·2-s + 729·3-s + 1.61e4·4-s − 5.21e4·5-s + 1.13e5·6-s − 1.21e5·7-s + 1.24e6·8-s + 5.31e5·9-s − 8.14e6·10-s − 5.92e6·11-s + 1.17e7·12-s + 3.41e7·13-s − 1.90e7·14-s − 3.80e7·15-s + 6.16e7·16-s + 1.67e8·17-s + 8.29e7·18-s − 1.20e8·19-s − 8.43e8·20-s − 8.88e7·21-s − 9.24e8·22-s − 8.89e8·23-s + 9.06e8·24-s + 1.50e9·25-s + 5.33e9·26-s + 3.87e8·27-s − 1.97e9·28-s + ⋯
L(s)  = 1  + 1.72·2-s + 0.577·3-s + 1.97·4-s − 1.49·5-s + 0.995·6-s − 0.391·7-s + 1.67·8-s + 0.333·9-s − 2.57·10-s − 1.00·11-s + 1.13·12-s + 1.96·13-s − 0.675·14-s − 0.862·15-s + 0.918·16-s + 1.68·17-s + 0.574·18-s − 0.588·19-s − 2.94·20-s − 0.226·21-s − 1.73·22-s − 1.25·23-s + 0.968·24-s + 1.23·25-s + 3.38·26-s + 0.192·27-s − 0.772·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(189.798\)
Root analytic conductor: \(13.7767\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 729T \)
59 \( 1 - 4.21e10T \)
good2 \( 1 - 156.T + 8.19e3T^{2} \)
5 \( 1 + 5.21e4T + 1.22e9T^{2} \)
7 \( 1 + 1.21e5T + 9.68e10T^{2} \)
11 \( 1 + 5.92e6T + 3.45e13T^{2} \)
13 \( 1 - 3.41e7T + 3.02e14T^{2} \)
17 \( 1 - 1.67e8T + 9.90e15T^{2} \)
19 \( 1 + 1.20e8T + 4.20e16T^{2} \)
23 \( 1 + 8.89e8T + 5.04e17T^{2} \)
29 \( 1 + 1.81e9T + 1.02e19T^{2} \)
31 \( 1 - 6.89e8T + 2.44e19T^{2} \)
37 \( 1 - 1.67e9T + 2.43e20T^{2} \)
41 \( 1 + 3.57e10T + 9.25e20T^{2} \)
43 \( 1 + 5.31e10T + 1.71e21T^{2} \)
47 \( 1 + 5.93e10T + 5.46e21T^{2} \)
53 \( 1 + 2.17e11T + 2.60e22T^{2} \)
61 \( 1 - 6.03e11T + 1.61e23T^{2} \)
67 \( 1 + 1.22e12T + 5.48e23T^{2} \)
71 \( 1 - 4.34e11T + 1.16e24T^{2} \)
73 \( 1 + 1.83e12T + 1.67e24T^{2} \)
79 \( 1 + 5.33e11T + 4.66e24T^{2} \)
83 \( 1 - 1.78e12T + 8.87e24T^{2} \)
89 \( 1 + 7.18e12T + 2.19e25T^{2} \)
97 \( 1 - 5.89e12T + 6.73e25T^{2} \)
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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

−10.20746009571667957986222030086, −8.358280342543167449911593776701, −7.79342659127237530924792237570, −6.56982583591856122596526785696, −5.56128051678031212740512499267, −4.34743842313661866232210298391, −3.47731302737099157325282459153, −3.19723435160871447160884742532, −1.61497462338893873569331882441, 0, 1.61497462338893873569331882441, 3.19723435160871447160884742532, 3.47731302737099157325282459153, 4.34743842313661866232210298391, 5.56128051678031212740512499267, 6.56982583591856122596526785696, 7.79342659127237530924792237570, 8.358280342543167449911593776701, 10.20746009571667957986222030086

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