Properties

Label 2-177-1.1-c13-0-14
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 170.·2-s + 729·3-s + 2.07e4·4-s + 7.03e3·5-s − 1.24e5·6-s − 1.66e5·7-s − 2.13e6·8-s + 5.31e5·9-s − 1.19e6·10-s − 2.55e6·11-s + 1.51e7·12-s − 9.38e6·13-s + 2.82e7·14-s + 5.12e6·15-s + 1.93e8·16-s − 4.94e7·17-s − 9.04e7·18-s + 8.01e6·19-s + 1.46e8·20-s − 1.21e8·21-s + 4.34e8·22-s − 4.03e8·23-s − 1.55e9·24-s − 1.17e9·25-s + 1.59e9·26-s + 3.87e8·27-s − 3.45e9·28-s + ⋯
L(s)  = 1  − 1.87·2-s + 0.577·3-s + 2.53·4-s + 0.201·5-s − 1.08·6-s − 0.534·7-s − 2.88·8-s + 0.333·9-s − 0.378·10-s − 0.434·11-s + 1.46·12-s − 0.539·13-s + 1.00·14-s + 0.116·15-s + 2.88·16-s − 0.497·17-s − 0.626·18-s + 0.0390·19-s + 0.510·20-s − 0.308·21-s + 0.817·22-s − 0.568·23-s − 1.66·24-s − 0.959·25-s + 1.01·26-s + 0.192·27-s − 1.35·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(0.4483468593\)
\(L(\frac12)\) \(\approx\) \(0.4483468593\)
\(L(\frac{15}{2})\) 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 \( 1 + 170.T + 8.19e3T^{2} \)
5 \( 1 - 7.03e3T + 1.22e9T^{2} \)
7 \( 1 + 1.66e5T + 9.68e10T^{2} \)
11 \( 1 + 2.55e6T + 3.45e13T^{2} \)
13 \( 1 + 9.38e6T + 3.02e14T^{2} \)
17 \( 1 + 4.94e7T + 9.90e15T^{2} \)
19 \( 1 - 8.01e6T + 4.20e16T^{2} \)
23 \( 1 + 4.03e8T + 5.04e17T^{2} \)
29 \( 1 + 1.78e9T + 1.02e19T^{2} \)
31 \( 1 + 7.59e9T + 2.44e19T^{2} \)
37 \( 1 + 2.95e10T + 2.43e20T^{2} \)
41 \( 1 + 1.30e9T + 9.25e20T^{2} \)
43 \( 1 - 1.57e9T + 1.71e21T^{2} \)
47 \( 1 - 1.19e11T + 5.46e21T^{2} \)
53 \( 1 - 6.19e10T + 2.60e22T^{2} \)
61 \( 1 - 4.65e11T + 1.61e23T^{2} \)
67 \( 1 - 8.10e11T + 5.48e23T^{2} \)
71 \( 1 + 2.60e11T + 1.16e24T^{2} \)
73 \( 1 - 4.55e11T + 1.67e24T^{2} \)
79 \( 1 + 3.34e11T + 4.66e24T^{2} \)
83 \( 1 + 3.25e12T + 8.87e24T^{2} \)
89 \( 1 + 5.57e11T + 2.19e25T^{2} \)
97 \( 1 - 8.44e12T + 6.73e25T^{2} \)
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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

−10.02031523269800578175709668770, −9.334577684490517800608463537549, −8.538747306194738871548801573917, −7.55632530397932970957782925358, −6.86970343762320752074821766789, −5.62647581280535717352977512012, −3.61072909201504769968036059072, −2.39295751055420313516262203802, −1.75894848815567349227572546997, −0.34817657305461631780348250968, 0.34817657305461631780348250968, 1.75894848815567349227572546997, 2.39295751055420313516262203802, 3.61072909201504769968036059072, 5.62647581280535717352977512012, 6.86970343762320752074821766789, 7.55632530397932970957782925358, 8.538747306194738871548801573917, 9.334577684490517800608463537549, 10.02031523269800578175709668770

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