Properties

Label 2-177-1.1-c13-0-42
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  − 138.·2-s + 729·3-s + 1.11e4·4-s − 4.14e4·5-s − 1.01e5·6-s − 7.09e4·7-s − 4.06e5·8-s + 5.31e5·9-s + 5.75e6·10-s − 5.18e6·11-s + 8.10e6·12-s + 2.69e7·13-s + 9.85e6·14-s − 3.02e7·15-s − 3.46e7·16-s + 1.85e8·17-s − 7.38e7·18-s + 3.25e8·19-s − 4.60e8·20-s − 5.17e7·21-s + 7.21e8·22-s + 1.17e9·23-s − 2.96e8·24-s + 4.96e8·25-s − 3.74e9·26-s + 3.87e8·27-s − 7.88e8·28-s + ⋯
L(s)  = 1  − 1.53·2-s + 0.577·3-s + 1.35·4-s − 1.18·5-s − 0.886·6-s − 0.227·7-s − 0.547·8-s + 0.333·9-s + 1.82·10-s − 0.883·11-s + 0.783·12-s + 1.54·13-s + 0.349·14-s − 0.684·15-s − 0.515·16-s + 1.86·17-s − 0.511·18-s + 1.58·19-s − 1.60·20-s − 0.131·21-s + 1.35·22-s + 1.65·23-s − 0.316·24-s + 0.406·25-s − 2.37·26-s + 0.192·27-s − 0.309·28-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

\(L(7)\) \(\approx\) \(1.278011532\)
\(L(\frac12)\) \(\approx\) \(1.278011532\)
\(L(\frac{15}{2})\) 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 \( 1 + 138.T + 8.19e3T^{2} \)
5 \( 1 + 4.14e4T + 1.22e9T^{2} \)
7 \( 1 + 7.09e4T + 9.68e10T^{2} \)
11 \( 1 + 5.18e6T + 3.45e13T^{2} \)
13 \( 1 - 2.69e7T + 3.02e14T^{2} \)
17 \( 1 - 1.85e8T + 9.90e15T^{2} \)
19 \( 1 - 3.25e8T + 4.20e16T^{2} \)
23 \( 1 - 1.17e9T + 5.04e17T^{2} \)
29 \( 1 - 2.93e9T + 1.02e19T^{2} \)
31 \( 1 + 1.23e9T + 2.44e19T^{2} \)
37 \( 1 + 1.79e10T + 2.43e20T^{2} \)
41 \( 1 - 5.41e10T + 9.25e20T^{2} \)
43 \( 1 + 4.68e10T + 1.71e21T^{2} \)
47 \( 1 - 2.60e10T + 5.46e21T^{2} \)
53 \( 1 - 1.43e11T + 2.60e22T^{2} \)
61 \( 1 - 5.85e11T + 1.61e23T^{2} \)
67 \( 1 + 1.43e11T + 5.48e23T^{2} \)
71 \( 1 - 9.28e11T + 1.16e24T^{2} \)
73 \( 1 - 1.49e12T + 1.67e24T^{2} \)
79 \( 1 + 2.13e12T + 4.66e24T^{2} \)
83 \( 1 - 5.02e11T + 8.87e24T^{2} \)
89 \( 1 - 2.89e12T + 2.19e25T^{2} \)
97 \( 1 - 1.54e12T + 6.73e25T^{2} \)
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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

−10.16220173305039005836113300469, −9.204503244279156176745495971252, −8.256039517619884354867381467236, −7.77606196901830788978770997556, −6.95001705202769056477116852038, −5.28161415200580693012777345605, −3.64522586255455427256054889920, −2.92724295976358617918704475104, −1.23588956117163306186426257773, −0.70946834517342866348090353233, 0.70946834517342866348090353233, 1.23588956117163306186426257773, 2.92724295976358617918704475104, 3.64522586255455427256054889920, 5.28161415200580693012777345605, 6.95001705202769056477116852038, 7.77606196901830788978770997556, 8.256039517619884354867381467236, 9.204503244279156176745495971252, 10.16220173305039005836113300469

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