Properties

Label 2-177-1.1-c11-0-44
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  − 10.8·2-s − 243·3-s − 1.92e3·4-s + 3.57e3·5-s + 2.64e3·6-s − 6.64e4·7-s + 4.33e4·8-s + 5.90e4·9-s − 3.89e4·10-s − 5.86e5·11-s + 4.68e5·12-s − 2.48e5·13-s + 7.24e5·14-s − 8.67e5·15-s + 3.47e6·16-s + 3.17e6·17-s − 6.43e5·18-s − 8.29e6·19-s − 6.88e6·20-s + 1.61e7·21-s + 6.38e6·22-s + 4.26e7·23-s − 1.05e7·24-s − 3.60e7·25-s + 2.71e6·26-s − 1.43e7·27-s + 1.28e8·28-s + ⋯
L(s)  = 1  − 0.240·2-s − 0.577·3-s − 0.941·4-s + 0.510·5-s + 0.139·6-s − 1.49·7-s + 0.467·8-s + 0.333·9-s − 0.123·10-s − 1.09·11-s + 0.543·12-s − 0.185·13-s + 0.360·14-s − 0.295·15-s + 0.829·16-s + 0.542·17-s − 0.0802·18-s − 0.768·19-s − 0.481·20-s + 0.863·21-s + 0.264·22-s + 1.38·23-s − 0.270·24-s − 0.738·25-s + 0.0447·26-s − 0.192·27-s + 1.40·28-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 \( 1 + 10.8T + 2.04e3T^{2} \)
5 \( 1 - 3.57e3T + 4.88e7T^{2} \)
7 \( 1 + 6.64e4T + 1.97e9T^{2} \)
11 \( 1 + 5.86e5T + 2.85e11T^{2} \)
13 \( 1 + 2.48e5T + 1.79e12T^{2} \)
17 \( 1 - 3.17e6T + 3.42e13T^{2} \)
19 \( 1 + 8.29e6T + 1.16e14T^{2} \)
23 \( 1 - 4.26e7T + 9.52e14T^{2} \)
29 \( 1 + 2.96e7T + 1.22e16T^{2} \)
31 \( 1 - 7.28e7T + 2.54e16T^{2} \)
37 \( 1 - 4.75e8T + 1.77e17T^{2} \)
41 \( 1 - 5.12e7T + 5.50e17T^{2} \)
43 \( 1 + 1.03e9T + 9.29e17T^{2} \)
47 \( 1 - 2.14e9T + 2.47e18T^{2} \)
53 \( 1 - 1.08e9T + 9.26e18T^{2} \)
61 \( 1 - 8.65e9T + 4.35e19T^{2} \)
67 \( 1 - 1.80e10T + 1.22e20T^{2} \)
71 \( 1 + 7.45e9T + 2.31e20T^{2} \)
73 \( 1 - 2.81e10T + 3.13e20T^{2} \)
79 \( 1 - 1.85e10T + 7.47e20T^{2} \)
83 \( 1 + 2.72e10T + 1.28e21T^{2} \)
89 \( 1 + 4.76e10T + 2.77e21T^{2} \)
97 \( 1 - 4.57e10T + 7.15e21T^{2} \)
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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

−9.951671406416369390173095578470, −9.496308173073160752095539449925, −8.256058427752288716172477228118, −7.01021554187658508019265453457, −5.89214631431764790727648474624, −5.08409075711198732399461682739, −3.78869663037892841241061932879, −2.55297829826557157886102812349, −0.852007744921931124803239668588, 0, 0.852007744921931124803239668588, 2.55297829826557157886102812349, 3.78869663037892841241061932879, 5.08409075711198732399461682739, 5.89214631431764790727648474624, 7.01021554187658508019265453457, 8.256058427752288716172477228118, 9.496308173073160752095539449925, 9.951671406416369390173095578470

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