Properties

Label 2-177-1.1-c13-0-36
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  − 1.12·2-s + 729·3-s − 8.19e3·4-s − 5.35e4·5-s − 819.·6-s + 4.45e5·7-s + 1.84e4·8-s + 5.31e5·9-s + 6.01e4·10-s + 4.07e6·11-s − 5.97e6·12-s + 2.06e7·13-s − 5.00e5·14-s − 3.90e7·15-s + 6.70e7·16-s + 1.55e7·17-s − 5.97e5·18-s − 3.30e8·19-s + 4.38e8·20-s + 3.24e8·21-s − 4.57e6·22-s − 3.80e8·23-s + 1.34e7·24-s + 1.64e9·25-s − 2.32e7·26-s + 3.87e8·27-s − 3.64e9·28-s + ⋯
L(s)  = 1  − 0.0124·2-s + 0.577·3-s − 0.999·4-s − 1.53·5-s − 0.00717·6-s + 1.43·7-s + 0.0248·8-s + 0.333·9-s + 0.0190·10-s + 0.692·11-s − 0.577·12-s + 1.18·13-s − 0.0177·14-s − 0.884·15-s + 0.999·16-s + 0.155·17-s − 0.00414·18-s − 1.61·19-s + 1.53·20-s + 0.825·21-s − 0.00860·22-s − 0.536·23-s + 0.0143·24-s + 1.34·25-s − 0.0147·26-s + 0.192·27-s − 1.43·28-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

\(L(7)\) \(\approx\) \(2.027025133\)
\(L(\frac12)\) \(\approx\) \(2.027025133\)
\(L(\frac{15}{2})\) 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 \( 1 + 1.12T + 8.19e3T^{2} \)
5 \( 1 + 5.35e4T + 1.22e9T^{2} \)
7 \( 1 - 4.45e5T + 9.68e10T^{2} \)
11 \( 1 - 4.07e6T + 3.45e13T^{2} \)
13 \( 1 - 2.06e7T + 3.02e14T^{2} \)
17 \( 1 - 1.55e7T + 9.90e15T^{2} \)
19 \( 1 + 3.30e8T + 4.20e16T^{2} \)
23 \( 1 + 3.80e8T + 5.04e17T^{2} \)
29 \( 1 - 3.38e9T + 1.02e19T^{2} \)
31 \( 1 - 6.98e9T + 2.44e19T^{2} \)
37 \( 1 - 8.29e9T + 2.43e20T^{2} \)
41 \( 1 - 4.02e10T + 9.25e20T^{2} \)
43 \( 1 + 5.64e10T + 1.71e21T^{2} \)
47 \( 1 + 1.39e11T + 5.46e21T^{2} \)
53 \( 1 - 3.09e10T + 2.60e22T^{2} \)
61 \( 1 + 6.44e11T + 1.61e23T^{2} \)
67 \( 1 + 1.03e12T + 5.48e23T^{2} \)
71 \( 1 - 1.62e12T + 1.16e24T^{2} \)
73 \( 1 + 8.01e11T + 1.67e24T^{2} \)
79 \( 1 + 1.43e12T + 4.66e24T^{2} \)
83 \( 1 - 1.93e12T + 8.87e24T^{2} \)
89 \( 1 - 1.58e12T + 2.19e25T^{2} \)
97 \( 1 - 9.46e12T + 6.73e25T^{2} \)
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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

−10.40523667689778635560768257566, −8.920259352649724187807573008835, −8.205139024055968794994286992073, −7.966515212243751178622983010183, −6.35731829607405998721405117618, −4.55333606111855261699123418064, −4.32352064498928415895499520152, −3.29834239903859786871067669772, −1.57724494958982126090911050886, −0.63853863441363294888857640900, 0.63853863441363294888857640900, 1.57724494958982126090911050886, 3.29834239903859786871067669772, 4.32352064498928415895499520152, 4.55333606111855261699123418064, 6.35731829607405998721405117618, 7.966515212243751178622983010183, 8.205139024055968794994286992073, 8.920259352649724187807573008835, 10.40523667689778635560768257566

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