Properties

Label 2-177-1.1-c11-0-28
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  − 81.8·2-s − 243·3-s + 4.65e3·4-s − 1.22e4·5-s + 1.98e4·6-s − 6.56e4·7-s − 2.13e5·8-s + 5.90e4·9-s + 1.00e6·10-s + 6.14e5·11-s − 1.13e6·12-s + 1.74e6·13-s + 5.37e6·14-s + 2.98e6·15-s + 7.96e6·16-s − 1.11e6·17-s − 4.83e6·18-s + 1.46e7·19-s − 5.71e7·20-s + 1.59e7·21-s − 5.03e7·22-s + 5.31e7·23-s + 5.19e7·24-s + 1.01e8·25-s − 1.42e8·26-s − 1.43e7·27-s − 3.05e8·28-s + ⋯
L(s)  = 1  − 1.80·2-s − 0.577·3-s + 2.27·4-s − 1.75·5-s + 1.04·6-s − 1.47·7-s − 2.30·8-s + 0.333·9-s + 3.17·10-s + 1.15·11-s − 1.31·12-s + 1.30·13-s + 2.67·14-s + 1.01·15-s + 1.89·16-s − 0.190·17-s − 0.603·18-s + 1.35·19-s − 3.99·20-s + 0.852·21-s − 2.08·22-s + 1.72·23-s + 1.33·24-s + 2.08·25-s − 2.35·26-s − 0.192·27-s − 3.35·28-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

\(L(6)\) \(\approx\) \(0.5488529215\)
\(L(\frac12)\) \(\approx\) \(0.5488529215\)
\(L(\frac{13}{2})\) 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 \( 1 + 81.8T + 2.04e3T^{2} \)
5 \( 1 + 1.22e4T + 4.88e7T^{2} \)
7 \( 1 + 6.56e4T + 1.97e9T^{2} \)
11 \( 1 - 6.14e5T + 2.85e11T^{2} \)
13 \( 1 - 1.74e6T + 1.79e12T^{2} \)
17 \( 1 + 1.11e6T + 3.42e13T^{2} \)
19 \( 1 - 1.46e7T + 1.16e14T^{2} \)
23 \( 1 - 5.31e7T + 9.52e14T^{2} \)
29 \( 1 + 1.59e8T + 1.22e16T^{2} \)
31 \( 1 - 4.50e7T + 2.54e16T^{2} \)
37 \( 1 - 7.33e8T + 1.77e17T^{2} \)
41 \( 1 + 3.90e7T + 5.50e17T^{2} \)
43 \( 1 - 9.67e8T + 9.29e17T^{2} \)
47 \( 1 - 1.84e9T + 2.47e18T^{2} \)
53 \( 1 - 3.64e9T + 9.26e18T^{2} \)
61 \( 1 - 1.84e9T + 4.35e19T^{2} \)
67 \( 1 + 1.36e10T + 1.22e20T^{2} \)
71 \( 1 + 2.07e10T + 2.31e20T^{2} \)
73 \( 1 - 1.69e10T + 3.13e20T^{2} \)
79 \( 1 - 3.36e10T + 7.47e20T^{2} \)
83 \( 1 - 2.76e10T + 1.28e21T^{2} \)
89 \( 1 + 1.44e10T + 2.77e21T^{2} \)
97 \( 1 - 1.18e11T + 7.15e21T^{2} \)
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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

−10.71937553934204423400915671369, −9.363962929619080383109142340955, −8.924193339147344982605314886847, −7.59308070403666729491555722559, −6.99904981112374460391625619057, −6.08823019388777552622614383595, −3.93163020422929902429844842239, −3.07639789861877699383917324318, −0.988528303858858589195725319409, −0.63233906518527973790064250575, 0.63233906518527973790064250575, 0.988528303858858589195725319409, 3.07639789861877699383917324318, 3.93163020422929902429844842239, 6.08823019388777552622614383595, 6.99904981112374460391625619057, 7.59308070403666729491555722559, 8.924193339147344982605314886847, 9.363962929619080383109142340955, 10.71937553934204423400915671369

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