Properties

Label 2-177-1.1-c11-0-74
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  + 60.1·2-s − 243·3-s + 1.57e3·4-s − 9.25e3·5-s − 1.46e4·6-s − 7.12e3·7-s − 2.85e4·8-s + 5.90e4·9-s − 5.57e5·10-s + 2.04e4·11-s − 3.82e5·12-s + 2.48e6·13-s − 4.28e5·14-s + 2.24e6·15-s − 4.94e6·16-s + 5.23e6·17-s + 3.55e6·18-s + 6.43e6·19-s − 1.45e7·20-s + 1.73e6·21-s + 1.22e6·22-s + 2.55e7·23-s + 6.93e6·24-s + 3.68e7·25-s + 1.49e8·26-s − 1.43e7·27-s − 1.12e7·28-s + ⋯
L(s)  = 1  + 1.32·2-s − 0.577·3-s + 0.768·4-s − 1.32·5-s − 0.767·6-s − 0.160·7-s − 0.307·8-s + 0.333·9-s − 1.76·10-s + 0.0382·11-s − 0.443·12-s + 1.85·13-s − 0.212·14-s + 0.764·15-s − 1.17·16-s + 0.893·17-s + 0.443·18-s + 0.595·19-s − 1.01·20-s + 0.0924·21-s + 0.0508·22-s + 0.829·23-s + 0.177·24-s + 0.755·25-s + 2.46·26-s − 0.192·27-s − 0.123·28-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 \( 1 - 60.1T + 2.04e3T^{2} \)
5 \( 1 + 9.25e3T + 4.88e7T^{2} \)
7 \( 1 + 7.12e3T + 1.97e9T^{2} \)
11 \( 1 - 2.04e4T + 2.85e11T^{2} \)
13 \( 1 - 2.48e6T + 1.79e12T^{2} \)
17 \( 1 - 5.23e6T + 3.42e13T^{2} \)
19 \( 1 - 6.43e6T + 1.16e14T^{2} \)
23 \( 1 - 2.55e7T + 9.52e14T^{2} \)
29 \( 1 + 1.12e8T + 1.22e16T^{2} \)
31 \( 1 - 1.37e8T + 2.54e16T^{2} \)
37 \( 1 + 6.05e8T + 1.77e17T^{2} \)
41 \( 1 + 3.64e8T + 5.50e17T^{2} \)
43 \( 1 + 4.14e8T + 9.29e17T^{2} \)
47 \( 1 - 2.42e9T + 2.47e18T^{2} \)
53 \( 1 + 6.03e9T + 9.26e18T^{2} \)
61 \( 1 + 4.98e9T + 4.35e19T^{2} \)
67 \( 1 - 1.57e10T + 1.22e20T^{2} \)
71 \( 1 + 2.25e10T + 2.31e20T^{2} \)
73 \( 1 + 9.52e9T + 3.13e20T^{2} \)
79 \( 1 + 2.72e10T + 7.47e20T^{2} \)
83 \( 1 - 5.80e10T + 1.28e21T^{2} \)
89 \( 1 - 1.01e11T + 2.77e21T^{2} \)
97 \( 1 - 4.67e10T + 7.15e21T^{2} \)
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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

−10.69381131306020577066031331762, −9.074884900658281219866393837263, −7.921575327668569885791646851599, −6.75578421089551345383665107286, −5.79358765807130889365315654356, −4.81967609883974333071895720913, −3.71715707824079507198615163864, −3.29014416365077368936844298644, −1.22944224924534102777109311462, 0, 1.22944224924534102777109311462, 3.29014416365077368936844298644, 3.71715707824079507198615163864, 4.81967609883974333071895720913, 5.79358765807130889365315654356, 6.75578421089551345383665107286, 7.921575327668569885791646851599, 9.074884900658281219866393837263, 10.69381131306020577066031331762

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