Properties

Label 2-177-1.1-c11-0-94
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 84.6·2-s + 243·3-s + 5.10e3·4-s + 8.36e3·5-s + 2.05e4·6-s − 4.17e4·7-s + 2.59e5·8-s + 5.90e4·9-s + 7.07e5·10-s + 8.33e5·11-s + 1.24e6·12-s + 2.33e6·13-s − 3.53e6·14-s + 2.03e6·15-s + 1.14e7·16-s − 3.86e6·17-s + 4.99e6·18-s + 1.20e6·19-s + 4.27e7·20-s − 1.01e7·21-s + 7.05e7·22-s − 4.06e7·23-s + 6.29e7·24-s + 2.11e7·25-s + 1.97e8·26-s + 1.43e7·27-s − 2.13e8·28-s + ⋯
L(s)  = 1  + 1.86·2-s + 0.577·3-s + 2.49·4-s + 1.19·5-s + 1.07·6-s − 0.938·7-s + 2.79·8-s + 0.333·9-s + 2.23·10-s + 1.56·11-s + 1.44·12-s + 1.74·13-s − 1.75·14-s + 0.690·15-s + 2.72·16-s − 0.660·17-s + 0.623·18-s + 0.111·19-s + 2.98·20-s − 0.541·21-s + 2.91·22-s − 1.31·23-s + 1.61·24-s + 0.432·25-s + 3.26·26-s + 0.192·27-s − 2.34·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(13.37208193\)
\(L(\frac12)\) \(\approx\) \(13.37208193\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 243T \)
59 \( 1 - 7.14e8T \)
good2 \( 1 - 84.6T + 2.04e3T^{2} \)
5 \( 1 - 8.36e3T + 4.88e7T^{2} \)
7 \( 1 + 4.17e4T + 1.97e9T^{2} \)
11 \( 1 - 8.33e5T + 2.85e11T^{2} \)
13 \( 1 - 2.33e6T + 1.79e12T^{2} \)
17 \( 1 + 3.86e6T + 3.42e13T^{2} \)
19 \( 1 - 1.20e6T + 1.16e14T^{2} \)
23 \( 1 + 4.06e7T + 9.52e14T^{2} \)
29 \( 1 + 4.76e7T + 1.22e16T^{2} \)
31 \( 1 + 1.88e8T + 2.54e16T^{2} \)
37 \( 1 + 3.93e8T + 1.77e17T^{2} \)
41 \( 1 - 2.32e8T + 5.50e17T^{2} \)
43 \( 1 + 1.33e9T + 9.29e17T^{2} \)
47 \( 1 - 2.07e9T + 2.47e18T^{2} \)
53 \( 1 - 3.09e9T + 9.26e18T^{2} \)
61 \( 1 - 8.26e9T + 4.35e19T^{2} \)
67 \( 1 - 1.60e10T + 1.22e20T^{2} \)
71 \( 1 + 1.52e9T + 2.31e20T^{2} \)
73 \( 1 + 3.03e10T + 3.13e20T^{2} \)
79 \( 1 - 4.55e10T + 7.47e20T^{2} \)
83 \( 1 + 2.27e9T + 1.28e21T^{2} \)
89 \( 1 - 2.81e10T + 2.77e21T^{2} \)
97 \( 1 + 1.58e10T + 7.15e21T^{2} \)
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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

−10.93418202203480493462248031785, −9.779896916025662055090762418696, −8.751444317014779933453976934102, −6.89194219301685588772806509205, −6.28894591468475252600470445439, −5.60041158409438495176810738326, −3.93064325561968688326837350303, −3.58253282474235117483350058794, −2.20672542187565831636493937699, −1.45776072540267242968261694205, 1.45776072540267242968261694205, 2.20672542187565831636493937699, 3.58253282474235117483350058794, 3.93064325561968688326837350303, 5.60041158409438495176810738326, 6.28894591468475252600470445439, 6.89194219301685588772806509205, 8.751444317014779933453976934102, 9.779896916025662055090762418696, 10.93418202203480493462248031785

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