Properties

Label 2-177-1.1-c13-0-25
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  + 68.2·2-s + 729·3-s − 3.53e3·4-s − 1.43e4·5-s + 4.97e4·6-s − 1.60e4·7-s − 8.00e5·8-s + 5.31e5·9-s − 9.76e5·10-s − 5.62e6·11-s − 2.57e6·12-s − 1.90e7·13-s − 1.09e6·14-s − 1.04e7·15-s − 2.56e7·16-s + 1.17e8·17-s + 3.62e7·18-s + 6.78e7·19-s + 5.06e7·20-s − 1.17e7·21-s − 3.84e8·22-s − 1.30e9·23-s − 5.83e8·24-s − 1.01e9·25-s − 1.29e9·26-s + 3.87e8·27-s + 5.68e7·28-s + ⋯
L(s)  = 1  + 0.753·2-s + 0.577·3-s − 0.431·4-s − 0.409·5-s + 0.435·6-s − 0.0516·7-s − 1.07·8-s + 0.333·9-s − 0.308·10-s − 0.958·11-s − 0.249·12-s − 1.09·13-s − 0.0389·14-s − 0.236·15-s − 0.381·16-s + 1.17·17-s + 0.251·18-s + 0.330·19-s + 0.176·20-s − 0.0298·21-s − 0.722·22-s − 1.83·23-s − 0.623·24-s − 0.832·25-s − 0.824·26-s + 0.192·27-s + 0.0223·28-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

\(L(7)\) \(\approx\) \(1.688420912\)
\(L(\frac12)\) \(\approx\) \(1.688420912\)
\(L(\frac{15}{2})\) 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 \( 1 - 68.2T + 8.19e3T^{2} \)
5 \( 1 + 1.43e4T + 1.22e9T^{2} \)
7 \( 1 + 1.60e4T + 9.68e10T^{2} \)
11 \( 1 + 5.62e6T + 3.45e13T^{2} \)
13 \( 1 + 1.90e7T + 3.02e14T^{2} \)
17 \( 1 - 1.17e8T + 9.90e15T^{2} \)
19 \( 1 - 6.78e7T + 4.20e16T^{2} \)
23 \( 1 + 1.30e9T + 5.04e17T^{2} \)
29 \( 1 - 1.88e8T + 1.02e19T^{2} \)
31 \( 1 + 4.39e9T + 2.44e19T^{2} \)
37 \( 1 - 2.15e10T + 2.43e20T^{2} \)
41 \( 1 - 7.66e9T + 9.25e20T^{2} \)
43 \( 1 - 4.14e10T + 1.71e21T^{2} \)
47 \( 1 + 1.24e11T + 5.46e21T^{2} \)
53 \( 1 - 1.16e11T + 2.60e22T^{2} \)
61 \( 1 - 9.07e10T + 1.61e23T^{2} \)
67 \( 1 - 2.79e11T + 5.48e23T^{2} \)
71 \( 1 - 8.96e11T + 1.16e24T^{2} \)
73 \( 1 - 2.19e9T + 1.67e24T^{2} \)
79 \( 1 + 2.36e12T + 4.66e24T^{2} \)
83 \( 1 - 3.89e12T + 8.87e24T^{2} \)
89 \( 1 - 3.38e12T + 2.19e25T^{2} \)
97 \( 1 + 2.54e12T + 6.73e25T^{2} \)
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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

−10.06039334454276774595397998365, −9.515174249133579438846817116679, −8.090667744145494612106079190093, −7.61613831676975017577637297857, −5.97830125279585497623223247544, −5.05821828985112681548125148160, −4.05492080812184401389542811215, −3.18008854408821501316206303166, −2.15775694019818570129910615695, −0.45372304043860494239553879120, 0.45372304043860494239553879120, 2.15775694019818570129910615695, 3.18008854408821501316206303166, 4.05492080812184401389542811215, 5.05821828985112681548125148160, 5.97830125279585497623223247544, 7.61613831676975017577637297857, 8.090667744145494612106079190093, 9.515174249133579438846817116679, 10.06039334454276774595397998365

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