Properties

Label 2-177-1.1-c11-0-20
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  + 79.4·2-s − 243·3-s + 4.27e3·4-s − 1.24e4·5-s − 1.93e4·6-s − 4.69e3·7-s + 1.76e5·8-s + 5.90e4·9-s − 9.93e5·10-s − 6.84e5·11-s − 1.03e6·12-s − 8.07e5·13-s − 3.72e5·14-s + 3.03e6·15-s + 5.30e6·16-s + 2.08e6·17-s + 4.69e6·18-s + 1.99e6·19-s − 5.33e7·20-s + 1.14e6·21-s − 5.44e7·22-s − 3.43e7·23-s − 4.29e7·24-s + 1.07e8·25-s − 6.41e7·26-s − 1.43e7·27-s − 2.00e7·28-s + ⋯
L(s)  = 1  + 1.75·2-s − 0.577·3-s + 2.08·4-s − 1.78·5-s − 1.01·6-s − 0.105·7-s + 1.90·8-s + 0.333·9-s − 3.14·10-s − 1.28·11-s − 1.20·12-s − 0.603·13-s − 0.185·14-s + 1.03·15-s + 1.26·16-s + 0.356·17-s + 0.585·18-s + 0.184·19-s − 3.72·20-s + 0.0609·21-s − 2.25·22-s − 1.11·23-s − 1.10·24-s + 2.19·25-s − 1.05·26-s − 0.192·27-s − 0.220·28-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

\(L(6)\) \(\approx\) \(2.636596841\)
\(L(\frac12)\) \(\approx\) \(2.636596841\)
\(L(\frac{13}{2})\) 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 \( 1 - 79.4T + 2.04e3T^{2} \)
5 \( 1 + 1.24e4T + 4.88e7T^{2} \)
7 \( 1 + 4.69e3T + 1.97e9T^{2} \)
11 \( 1 + 6.84e5T + 2.85e11T^{2} \)
13 \( 1 + 8.07e5T + 1.79e12T^{2} \)
17 \( 1 - 2.08e6T + 3.42e13T^{2} \)
19 \( 1 - 1.99e6T + 1.16e14T^{2} \)
23 \( 1 + 3.43e7T + 9.52e14T^{2} \)
29 \( 1 - 5.13e7T + 1.22e16T^{2} \)
31 \( 1 - 2.77e8T + 2.54e16T^{2} \)
37 \( 1 + 1.29e8T + 1.77e17T^{2} \)
41 \( 1 - 4.27e8T + 5.50e17T^{2} \)
43 \( 1 - 1.12e9T + 9.29e17T^{2} \)
47 \( 1 + 1.50e9T + 2.47e18T^{2} \)
53 \( 1 - 3.00e9T + 9.26e18T^{2} \)
61 \( 1 - 5.99e9T + 4.35e19T^{2} \)
67 \( 1 + 1.44e10T + 1.22e20T^{2} \)
71 \( 1 - 1.55e10T + 2.31e20T^{2} \)
73 \( 1 + 2.93e9T + 3.13e20T^{2} \)
79 \( 1 - 4.42e10T + 7.47e20T^{2} \)
83 \( 1 - 1.50e10T + 1.28e21T^{2} \)
89 \( 1 - 5.19e10T + 2.77e21T^{2} \)
97 \( 1 - 1.11e11T + 7.15e21T^{2} \)
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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

−11.16586030047851522220662359105, −10.23651531411283158252816838015, −8.065901133686287596419257512283, −7.40122053894010050998626377703, −6.29394747253648945256429109777, −5.09047001651599706589530692188, −4.45568027927622799089214349223, −3.48440654209330663196167904457, −2.51200457747323447328152785521, −0.55215875632420820730457723423, 0.55215875632420820730457723423, 2.51200457747323447328152785521, 3.48440654209330663196167904457, 4.45568027927622799089214349223, 5.09047001651599706589530692188, 6.29394747253648945256429109777, 7.40122053894010050998626377703, 8.065901133686287596419257512283, 10.23651531411283158252816838015, 11.16586030047851522220662359105

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