Properties

Label 2-177-1.1-c9-0-42
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 22.9·2-s − 81·3-s + 13.3·4-s + 1.89e3·5-s + 1.85e3·6-s − 1.05e4·7-s + 1.14e4·8-s + 6.56e3·9-s − 4.33e4·10-s − 1.13e4·11-s − 1.08e3·12-s + 7.51e4·13-s + 2.42e5·14-s − 1.53e5·15-s − 2.68e5·16-s + 1.00e4·17-s − 1.50e5·18-s + 1.90e5·19-s + 2.52e4·20-s + 8.57e5·21-s + 2.59e5·22-s − 1.70e6·23-s − 9.25e5·24-s + 1.62e6·25-s − 1.72e6·26-s − 5.31e5·27-s − 1.41e5·28-s + ⋯
L(s)  = 1  − 1.01·2-s − 0.577·3-s + 0.0260·4-s + 1.35·5-s + 0.584·6-s − 1.66·7-s + 0.986·8-s + 0.333·9-s − 1.37·10-s − 0.232·11-s − 0.0150·12-s + 0.730·13-s + 1.68·14-s − 0.781·15-s − 1.02·16-s + 0.0291·17-s − 0.337·18-s + 0.334·19-s + 0.0352·20-s + 0.962·21-s + 0.235·22-s − 1.26·23-s − 0.569·24-s + 0.833·25-s − 0.739·26-s − 0.192·27-s − 0.0434·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 + 22.9T + 512T^{2} \)
5 \( 1 - 1.89e3T + 1.95e6T^{2} \)
7 \( 1 + 1.05e4T + 4.03e7T^{2} \)
11 \( 1 + 1.13e4T + 2.35e9T^{2} \)
13 \( 1 - 7.51e4T + 1.06e10T^{2} \)
17 \( 1 - 1.00e4T + 1.18e11T^{2} \)
19 \( 1 - 1.90e5T + 3.22e11T^{2} \)
23 \( 1 + 1.70e6T + 1.80e12T^{2} \)
29 \( 1 + 4.88e6T + 1.45e13T^{2} \)
31 \( 1 - 1.15e6T + 2.64e13T^{2} \)
37 \( 1 - 1.23e7T + 1.29e14T^{2} \)
41 \( 1 - 2.65e7T + 3.27e14T^{2} \)
43 \( 1 + 3.02e6T + 5.02e14T^{2} \)
47 \( 1 - 1.63e7T + 1.11e15T^{2} \)
53 \( 1 - 8.99e7T + 3.29e15T^{2} \)
61 \( 1 + 7.84e6T + 1.16e16T^{2} \)
67 \( 1 + 1.02e7T + 2.72e16T^{2} \)
71 \( 1 - 7.77e7T + 4.58e16T^{2} \)
73 \( 1 - 9.47e7T + 5.88e16T^{2} \)
79 \( 1 - 2.02e8T + 1.19e17T^{2} \)
83 \( 1 + 5.81e8T + 1.86e17T^{2} \)
89 \( 1 + 1.16e9T + 3.50e17T^{2} \)
97 \( 1 + 1.43e9T + 7.60e17T^{2} \)
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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

−10.02277524605458696670291320002, −9.754714761741650098594924278585, −8.872011744887449951000002554388, −7.42718101554420397282904415326, −6.24838475783749980051538961035, −5.64672956948037015178709096416, −3.95172272596931630068909492844, −2.33933660827455676232811864320, −1.05936726484483180731019567642, 0, 1.05936726484483180731019567642, 2.33933660827455676232811864320, 3.95172272596931630068909492844, 5.64672956948037015178709096416, 6.24838475783749980051538961035, 7.42718101554420397282904415326, 8.872011744887449951000002554388, 9.754714761741650098594924278585, 10.02277524605458696670291320002

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