Properties

Label 2-177-1.1-c13-0-109
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  + 32.0·2-s − 729·3-s − 7.16e3·4-s + 5.23e4·5-s − 2.33e4·6-s + 4.85e5·7-s − 4.91e5·8-s + 5.31e5·9-s + 1.67e6·10-s + 7.71e6·11-s + 5.22e6·12-s − 1.68e7·13-s + 1.55e7·14-s − 3.81e7·15-s + 4.29e7·16-s + 1.67e7·17-s + 1.70e7·18-s − 1.60e8·19-s − 3.75e8·20-s − 3.54e8·21-s + 2.47e8·22-s − 7.81e8·23-s + 3.58e8·24-s + 1.51e9·25-s − 5.39e8·26-s − 3.87e8·27-s − 3.48e9·28-s + ⋯
L(s)  = 1  + 0.353·2-s − 0.577·3-s − 0.874·4-s + 1.49·5-s − 0.204·6-s + 1.56·7-s − 0.663·8-s + 0.333·9-s + 0.530·10-s + 1.31·11-s + 0.505·12-s − 0.968·13-s + 0.552·14-s − 0.865·15-s + 0.639·16-s + 0.167·17-s + 0.117·18-s − 0.783·19-s − 1.31·20-s − 0.900·21-s + 0.464·22-s − 1.10·23-s + 0.383·24-s + 1.24·25-s − 0.342·26-s − 0.192·27-s − 1.36·28-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 \( 1 - 32.0T + 8.19e3T^{2} \)
5 \( 1 - 5.23e4T + 1.22e9T^{2} \)
7 \( 1 - 4.85e5T + 9.68e10T^{2} \)
11 \( 1 - 7.71e6T + 3.45e13T^{2} \)
13 \( 1 + 1.68e7T + 3.02e14T^{2} \)
17 \( 1 - 1.67e7T + 9.90e15T^{2} \)
19 \( 1 + 1.60e8T + 4.20e16T^{2} \)
23 \( 1 + 7.81e8T + 5.04e17T^{2} \)
29 \( 1 + 4.88e9T + 1.02e19T^{2} \)
31 \( 1 + 5.16e9T + 2.44e19T^{2} \)
37 \( 1 + 1.18e10T + 2.43e20T^{2} \)
41 \( 1 - 3.96e10T + 9.25e20T^{2} \)
43 \( 1 + 4.22e10T + 1.71e21T^{2} \)
47 \( 1 + 1.14e11T + 5.46e21T^{2} \)
53 \( 1 + 1.15e11T + 2.60e22T^{2} \)
61 \( 1 + 1.57e11T + 1.61e23T^{2} \)
67 \( 1 + 1.15e12T + 5.48e23T^{2} \)
71 \( 1 - 1.50e12T + 1.16e24T^{2} \)
73 \( 1 - 1.68e12T + 1.67e24T^{2} \)
79 \( 1 - 1.28e12T + 4.66e24T^{2} \)
83 \( 1 - 3.33e12T + 8.87e24T^{2} \)
89 \( 1 + 6.70e12T + 2.19e25T^{2} \)
97 \( 1 + 4.23e12T + 6.73e25T^{2} \)
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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

−9.714895755006858644465279579470, −9.118868623533122131636112546971, −7.891779536665510045589031618211, −6.44645213197205897821933452544, −5.52188021889494680954536621656, −4.88012258683499332352862392411, −3.90122938364644030803919484751, −2.00186954362490139581055029386, −1.44229553885386703697605895317, 0, 1.44229553885386703697605895317, 2.00186954362490139581055029386, 3.90122938364644030803919484751, 4.88012258683499332352862392411, 5.52188021889494680954536621656, 6.44645213197205897821933452544, 7.891779536665510045589031618211, 9.118868623533122131636112546971, 9.714895755006858644465279579470

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