Properties

Label 2-177-1.1-c11-0-8
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  + 23.0·2-s + 243·3-s − 1.51e3·4-s + 32.6·5-s + 5.61e3·6-s − 8.08e4·7-s − 8.22e4·8-s + 5.90e4·9-s + 754.·10-s − 1.96e5·11-s − 3.68e5·12-s − 2.11e6·13-s − 1.86e6·14-s + 7.94e3·15-s + 1.20e6·16-s − 4.62e6·17-s + 1.36e6·18-s + 4.04e6·19-s − 4.94e4·20-s − 1.96e7·21-s − 4.54e6·22-s − 2.90e7·23-s − 1.99e7·24-s − 4.88e7·25-s − 4.88e7·26-s + 1.43e7·27-s + 1.22e8·28-s + ⋯
L(s)  = 1  + 0.510·2-s + 0.577·3-s − 0.739·4-s + 0.00467·5-s + 0.294·6-s − 1.81·7-s − 0.887·8-s + 0.333·9-s + 0.00238·10-s − 0.368·11-s − 0.427·12-s − 1.58·13-s − 0.927·14-s + 0.00269·15-s + 0.286·16-s − 0.789·17-s + 0.170·18-s + 0.374·19-s − 0.00345·20-s − 1.04·21-s − 0.187·22-s − 0.942·23-s − 0.512·24-s − 0.999·25-s − 0.806·26-s + 0.192·27-s + 1.34·28-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

\(L(6)\) \(\approx\) \(0.4537642219\)
\(L(\frac12)\) \(\approx\) \(0.4537642219\)
\(L(\frac{13}{2})\) 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 \( 1 - 23.0T + 2.04e3T^{2} \)
5 \( 1 - 32.6T + 4.88e7T^{2} \)
7 \( 1 + 8.08e4T + 1.97e9T^{2} \)
11 \( 1 + 1.96e5T + 2.85e11T^{2} \)
13 \( 1 + 2.11e6T + 1.79e12T^{2} \)
17 \( 1 + 4.62e6T + 3.42e13T^{2} \)
19 \( 1 - 4.04e6T + 1.16e14T^{2} \)
23 \( 1 + 2.90e7T + 9.52e14T^{2} \)
29 \( 1 - 1.75e8T + 1.22e16T^{2} \)
31 \( 1 + 2.81e8T + 2.54e16T^{2} \)
37 \( 1 + 3.05e8T + 1.77e17T^{2} \)
41 \( 1 + 1.04e9T + 5.50e17T^{2} \)
43 \( 1 - 8.82e8T + 9.29e17T^{2} \)
47 \( 1 - 2.35e9T + 2.47e18T^{2} \)
53 \( 1 + 2.72e9T + 9.26e18T^{2} \)
61 \( 1 + 7.46e9T + 4.35e19T^{2} \)
67 \( 1 + 1.03e9T + 1.22e20T^{2} \)
71 \( 1 - 2.41e9T + 2.31e20T^{2} \)
73 \( 1 - 2.86e10T + 3.13e20T^{2} \)
79 \( 1 - 6.54e9T + 7.47e20T^{2} \)
83 \( 1 + 5.44e10T + 1.28e21T^{2} \)
89 \( 1 - 4.77e10T + 2.77e21T^{2} \)
97 \( 1 - 6.19e10T + 7.15e21T^{2} \)
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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

−10.25404785299841933434573703703, −9.629553292671776976505026619562, −8.908276710164619254922356086434, −7.56887390117487366974191227207, −6.49125533726119618575286192694, −5.35473758333202429819142041556, −4.15812717600535827624152507332, −3.24376365420593909954272852762, −2.34162627639989883383467787237, −0.25476641786243324156357562424, 0.25476641786243324156357562424, 2.34162627639989883383467787237, 3.24376365420593909954272852762, 4.15812717600535827624152507332, 5.35473758333202429819142041556, 6.49125533726119618575286192694, 7.56887390117487366974191227207, 8.908276710164619254922356086434, 9.629553292671776976505026619562, 10.25404785299841933434573703703

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