Properties

Label 2-177-1.1-c11-0-95
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  − 78.8·2-s + 243·3-s + 4.16e3·4-s + 6.89e3·5-s − 1.91e4·6-s + 4.11e4·7-s − 1.66e5·8-s + 5.90e4·9-s − 5.43e5·10-s − 4.69e5·11-s + 1.01e6·12-s + 1.92e6·13-s − 3.24e6·14-s + 1.67e6·15-s + 4.61e6·16-s − 6.53e6·17-s − 4.65e6·18-s + 1.40e7·19-s + 2.87e7·20-s + 9.99e6·21-s + 3.70e7·22-s + 4.11e6·23-s − 4.05e7·24-s − 1.29e6·25-s − 1.52e8·26-s + 1.43e7·27-s + 1.71e8·28-s + ⋯
L(s)  = 1  − 1.74·2-s + 0.577·3-s + 2.03·4-s + 0.986·5-s − 1.00·6-s + 0.924·7-s − 1.79·8-s + 0.333·9-s − 1.71·10-s − 0.879·11-s + 1.17·12-s + 1.44·13-s − 1.61·14-s + 0.569·15-s + 1.10·16-s − 1.11·17-s − 0.580·18-s + 1.30·19-s + 2.00·20-s + 0.534·21-s + 1.53·22-s + 0.133·23-s − 1.03·24-s − 0.0266·25-s − 2.50·26-s + 0.192·27-s + 1.88·28-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 \( 1 + 78.8T + 2.04e3T^{2} \)
5 \( 1 - 6.89e3T + 4.88e7T^{2} \)
7 \( 1 - 4.11e4T + 1.97e9T^{2} \)
11 \( 1 + 4.69e5T + 2.85e11T^{2} \)
13 \( 1 - 1.92e6T + 1.79e12T^{2} \)
17 \( 1 + 6.53e6T + 3.42e13T^{2} \)
19 \( 1 - 1.40e7T + 1.16e14T^{2} \)
23 \( 1 - 4.11e6T + 9.52e14T^{2} \)
29 \( 1 + 1.20e8T + 1.22e16T^{2} \)
31 \( 1 + 6.86e7T + 2.54e16T^{2} \)
37 \( 1 + 4.59e8T + 1.77e17T^{2} \)
41 \( 1 + 9.53e8T + 5.50e17T^{2} \)
43 \( 1 + 1.75e9T + 9.29e17T^{2} \)
47 \( 1 + 7.20e8T + 2.47e18T^{2} \)
53 \( 1 + 4.93e9T + 9.26e18T^{2} \)
61 \( 1 + 6.74e9T + 4.35e19T^{2} \)
67 \( 1 + 8.57e9T + 1.22e20T^{2} \)
71 \( 1 - 3.87e8T + 2.31e20T^{2} \)
73 \( 1 + 6.36e9T + 3.13e20T^{2} \)
79 \( 1 - 9.47e9T + 7.47e20T^{2} \)
83 \( 1 - 2.58e10T + 1.28e21T^{2} \)
89 \( 1 + 5.07e10T + 2.77e21T^{2} \)
97 \( 1 - 2.09e10T + 7.15e21T^{2} \)
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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

−9.953936215348964811080395296947, −9.070574176346674680398265782096, −8.376640584506696967431315844119, −7.55469441951552112625036457567, −6.42054506552825795329115919217, −5.13981590624280698162707146451, −3.19203274865512287626602272819, −1.79091505268608845254314627172, −1.56430324487283690159848899121, 0, 1.56430324487283690159848899121, 1.79091505268608845254314627172, 3.19203274865512287626602272819, 5.13981590624280698162707146451, 6.42054506552825795329115919217, 7.55469441951552112625036457567, 8.376640584506696967431315844119, 9.070574176346674680398265782096, 9.953936215348964811080395296947

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