Properties

Label 2-177-1.1-c11-0-36
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  − 42.1·2-s − 243·3-s − 273.·4-s − 6.33e3·5-s + 1.02e4·6-s − 2.29e4·7-s + 9.77e4·8-s + 5.90e4·9-s + 2.67e5·10-s − 4.34e4·11-s + 6.64e4·12-s − 8.20e5·13-s + 9.66e5·14-s + 1.54e6·15-s − 3.55e6·16-s − 5.29e6·17-s − 2.48e6·18-s − 9.56e6·19-s + 1.73e6·20-s + 5.57e6·21-s + 1.83e6·22-s − 1.56e7·23-s − 2.37e7·24-s − 8.64e6·25-s + 3.45e7·26-s − 1.43e7·27-s + 6.27e6·28-s + ⋯
L(s)  = 1  − 0.930·2-s − 0.577·3-s − 0.133·4-s − 0.907·5-s + 0.537·6-s − 0.516·7-s + 1.05·8-s + 0.333·9-s + 0.844·10-s − 0.0813·11-s + 0.0770·12-s − 0.612·13-s + 0.480·14-s + 0.523·15-s − 0.848·16-s − 0.903·17-s − 0.310·18-s − 0.885·19-s + 0.121·20-s + 0.297·21-s + 0.0757·22-s − 0.506·23-s − 0.609·24-s − 0.177·25-s + 0.570·26-s − 0.192·27-s + 0.0688·28-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 \( 1 + 42.1T + 2.04e3T^{2} \)
5 \( 1 + 6.33e3T + 4.88e7T^{2} \)
7 \( 1 + 2.29e4T + 1.97e9T^{2} \)
11 \( 1 + 4.34e4T + 2.85e11T^{2} \)
13 \( 1 + 8.20e5T + 1.79e12T^{2} \)
17 \( 1 + 5.29e6T + 3.42e13T^{2} \)
19 \( 1 + 9.56e6T + 1.16e14T^{2} \)
23 \( 1 + 1.56e7T + 9.52e14T^{2} \)
29 \( 1 + 4.39e7T + 1.22e16T^{2} \)
31 \( 1 - 2.36e8T + 2.54e16T^{2} \)
37 \( 1 - 4.01e8T + 1.77e17T^{2} \)
41 \( 1 + 8.87e8T + 5.50e17T^{2} \)
43 \( 1 - 1.42e9T + 9.29e17T^{2} \)
47 \( 1 - 3.01e9T + 2.47e18T^{2} \)
53 \( 1 + 1.28e8T + 9.26e18T^{2} \)
61 \( 1 + 9.01e8T + 4.35e19T^{2} \)
67 \( 1 + 4.70e9T + 1.22e20T^{2} \)
71 \( 1 - 2.04e10T + 2.31e20T^{2} \)
73 \( 1 + 1.87e10T + 3.13e20T^{2} \)
79 \( 1 - 4.04e10T + 7.47e20T^{2} \)
83 \( 1 - 5.18e10T + 1.28e21T^{2} \)
89 \( 1 + 2.60e10T + 2.77e21T^{2} \)
97 \( 1 + 1.89e10T + 7.15e21T^{2} \)
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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

−10.13559634327074986162327877079, −9.191932689403741659496742061331, −8.170662102703397906913760564651, −7.33710104728801336876617976949, −6.25122914013319362300856633283, −4.72242025553331719499481936445, −3.96301640038694177597069798105, −2.25071802997293674759371326046, −0.71563742253044930249533235300, 0, 0.71563742253044930249533235300, 2.25071802997293674759371326046, 3.96301640038694177597069798105, 4.72242025553331719499481936445, 6.25122914013319362300856633283, 7.33710104728801336876617976949, 8.170662102703397906913760564651, 9.191932689403741659496742061331, 10.13559634327074986162327877079

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