Properties

Label 2-177-1.1-c11-0-86
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 45.6·2-s + 243·3-s + 35.8·4-s − 2.29e3·5-s − 1.10e4·6-s + 5.53e4·7-s + 9.18e4·8-s + 5.90e4·9-s + 1.04e5·10-s + 9.46e5·11-s + 8.71e3·12-s + 8.53e5·13-s − 2.52e6·14-s − 5.57e5·15-s − 4.26e6·16-s − 1.00e7·17-s − 2.69e6·18-s − 6.49e6·19-s − 8.21e4·20-s + 1.34e7·21-s − 4.32e7·22-s − 3.83e7·23-s + 2.23e7·24-s − 4.35e7·25-s − 3.89e7·26-s + 1.43e7·27-s + 1.98e6·28-s + ⋯
L(s)  = 1  − 1.00·2-s + 0.577·3-s + 0.0175·4-s − 0.328·5-s − 0.582·6-s + 1.24·7-s + 0.991·8-s + 0.333·9-s + 0.330·10-s + 1.77·11-s + 0.0101·12-s + 0.637·13-s − 1.25·14-s − 0.189·15-s − 1.01·16-s − 1.71·17-s − 0.336·18-s − 0.601·19-s − 0.00574·20-s + 0.719·21-s − 1.78·22-s − 1.24·23-s + 0.572·24-s − 0.892·25-s − 0.643·26-s + 0.192·27-s + 0.0217·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) 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 \( 1 + 45.6T + 2.04e3T^{2} \)
5 \( 1 + 2.29e3T + 4.88e7T^{2} \)
7 \( 1 - 5.53e4T + 1.97e9T^{2} \)
11 \( 1 - 9.46e5T + 2.85e11T^{2} \)
13 \( 1 - 8.53e5T + 1.79e12T^{2} \)
17 \( 1 + 1.00e7T + 3.42e13T^{2} \)
19 \( 1 + 6.49e6T + 1.16e14T^{2} \)
23 \( 1 + 3.83e7T + 9.52e14T^{2} \)
29 \( 1 - 8.96e7T + 1.22e16T^{2} \)
31 \( 1 + 1.71e8T + 2.54e16T^{2} \)
37 \( 1 - 4.13e8T + 1.77e17T^{2} \)
41 \( 1 + 1.94e8T + 5.50e17T^{2} \)
43 \( 1 + 4.76e8T + 9.29e17T^{2} \)
47 \( 1 + 1.06e9T + 2.47e18T^{2} \)
53 \( 1 + 3.74e8T + 9.26e18T^{2} \)
61 \( 1 - 8.62e9T + 4.35e19T^{2} \)
67 \( 1 + 2.01e10T + 1.22e20T^{2} \)
71 \( 1 + 8.96e9T + 2.31e20T^{2} \)
73 \( 1 + 1.78e10T + 3.13e20T^{2} \)
79 \( 1 + 1.08e10T + 7.47e20T^{2} \)
83 \( 1 + 5.12e10T + 1.28e21T^{2} \)
89 \( 1 - 2.09e9T + 2.77e21T^{2} \)
97 \( 1 + 8.78e10T + 7.15e21T^{2} \)
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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

−9.926312805787545326232406160928, −8.813026105278598749485065420127, −8.521509377066003660547234400542, −7.49651649431388693564220161452, −6.35474962903804271254134375616, −4.46025910305863248336756939355, −3.99663886327474887712910188635, −1.95896984532962195543962447056, −1.37143144765841662447824040679, 0, 1.37143144765841662447824040679, 1.95896984532962195543962447056, 3.99663886327474887712910188635, 4.46025910305863248336756939355, 6.35474962903804271254134375616, 7.49651649431388693564220161452, 8.521509377066003660547234400542, 8.813026105278598749485065420127, 9.926312805787545326232406160928

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