Properties

Label 2-177-1.1-c9-0-10
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 14.9·2-s + 81·3-s − 288.·4-s − 659.·5-s + 1.21e3·6-s − 8.23e3·7-s − 1.19e4·8-s + 6.56e3·9-s − 9.84e3·10-s − 8.61e4·11-s − 2.33e4·12-s + 4.37e4·13-s − 1.22e5·14-s − 5.33e4·15-s − 3.08e4·16-s − 3.93e5·17-s + 9.80e4·18-s + 1.66e5·19-s + 1.90e5·20-s − 6.66e5·21-s − 1.28e6·22-s + 1.77e6·23-s − 9.69e5·24-s − 1.51e6·25-s + 6.54e5·26-s + 5.31e5·27-s + 2.37e6·28-s + ⋯
L(s)  = 1  + 0.660·2-s + 0.577·3-s − 0.564·4-s − 0.471·5-s + 0.381·6-s − 1.29·7-s − 1.03·8-s + 0.333·9-s − 0.311·10-s − 1.77·11-s − 0.325·12-s + 0.425·13-s − 0.855·14-s − 0.272·15-s − 0.117·16-s − 1.14·17-s + 0.220·18-s + 0.293·19-s + 0.266·20-s − 0.748·21-s − 1.17·22-s + 1.31·23-s − 0.596·24-s − 0.777·25-s + 0.280·26-s + 0.192·27-s + 0.730·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(1.110067941\)
\(L(\frac12)\) \(\approx\) \(1.110067941\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 81T \)
59 \( 1 + 1.21e7T \)
good2 \( 1 - 14.9T + 512T^{2} \)
5 \( 1 + 659.T + 1.95e6T^{2} \)
7 \( 1 + 8.23e3T + 4.03e7T^{2} \)
11 \( 1 + 8.61e4T + 2.35e9T^{2} \)
13 \( 1 - 4.37e4T + 1.06e10T^{2} \)
17 \( 1 + 3.93e5T + 1.18e11T^{2} \)
19 \( 1 - 1.66e5T + 3.22e11T^{2} \)
23 \( 1 - 1.77e6T + 1.80e12T^{2} \)
29 \( 1 - 3.21e6T + 1.45e13T^{2} \)
31 \( 1 + 2.76e6T + 2.64e13T^{2} \)
37 \( 1 + 4.27e5T + 1.29e14T^{2} \)
41 \( 1 - 1.68e7T + 3.27e14T^{2} \)
43 \( 1 - 2.06e5T + 5.02e14T^{2} \)
47 \( 1 + 1.61e7T + 1.11e15T^{2} \)
53 \( 1 + 8.10e6T + 3.29e15T^{2} \)
61 \( 1 - 3.93e7T + 1.16e16T^{2} \)
67 \( 1 - 3.07e8T + 2.72e16T^{2} \)
71 \( 1 + 2.50e8T + 4.58e16T^{2} \)
73 \( 1 + 1.16e8T + 5.88e16T^{2} \)
79 \( 1 + 3.26e8T + 1.19e17T^{2} \)
83 \( 1 - 4.90e8T + 1.86e17T^{2} \)
89 \( 1 + 6.22e8T + 3.50e17T^{2} \)
97 \( 1 - 6.21e8T + 7.60e17T^{2} \)
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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

−11.01133340899773673177993057880, −9.888174918626022866202988389465, −9.000946181094382858353050553449, −8.047054774480507072319408658808, −6.81724072539907511790934537360, −5.57949641772379708823834234897, −4.45644604331332679078500128102, −3.35503295797928115358570291559, −2.63212822110591396833912864996, −0.43195098339386381479702676204, 0.43195098339386381479702676204, 2.63212822110591396833912864996, 3.35503295797928115358570291559, 4.45644604331332679078500128102, 5.57949641772379708823834234897, 6.81724072539907511790934537360, 8.047054774480507072319408658808, 9.000946181094382858353050553449, 9.888174918626022866202988389465, 11.01133340899773673177993057880

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