Properties

Label 2-177-1.1-c11-0-32
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  − 39.1·2-s + 243·3-s − 515.·4-s + 158.·5-s − 9.51e3·6-s − 7.01e4·7-s + 1.00e5·8-s + 5.90e4·9-s − 6.21e3·10-s + 4.92e5·11-s − 1.25e5·12-s + 2.38e6·13-s + 2.74e6·14-s + 3.85e4·15-s − 2.87e6·16-s + 4.34e6·17-s − 2.31e6·18-s + 1.74e7·19-s − 8.18e4·20-s − 1.70e7·21-s − 1.92e7·22-s + 3.13e6·23-s + 2.43e7·24-s − 4.88e7·25-s − 9.35e7·26-s + 1.43e7·27-s + 3.61e7·28-s + ⋯
L(s)  = 1  − 0.864·2-s + 0.577·3-s − 0.251·4-s + 0.0227·5-s − 0.499·6-s − 1.57·7-s + 1.08·8-s + 0.333·9-s − 0.0196·10-s + 0.921·11-s − 0.145·12-s + 1.78·13-s + 1.36·14-s + 0.0131·15-s − 0.684·16-s + 0.742·17-s − 0.288·18-s + 1.62·19-s − 0.00572·20-s − 0.910·21-s − 0.797·22-s + 0.101·23-s + 0.625·24-s − 0.999·25-s − 1.54·26-s + 0.192·27-s + 0.397·28-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

\(L(6)\) \(\approx\) \(1.522968775\)
\(L(\frac12)\) \(\approx\) \(1.522968775\)
\(L(\frac{13}{2})\) 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 \( 1 + 39.1T + 2.04e3T^{2} \)
5 \( 1 - 158.T + 4.88e7T^{2} \)
7 \( 1 + 7.01e4T + 1.97e9T^{2} \)
11 \( 1 - 4.92e5T + 2.85e11T^{2} \)
13 \( 1 - 2.38e6T + 1.79e12T^{2} \)
17 \( 1 - 4.34e6T + 3.42e13T^{2} \)
19 \( 1 - 1.74e7T + 1.16e14T^{2} \)
23 \( 1 - 3.13e6T + 9.52e14T^{2} \)
29 \( 1 + 4.79e7T + 1.22e16T^{2} \)
31 \( 1 + 6.76e7T + 2.54e16T^{2} \)
37 \( 1 + 2.35e8T + 1.77e17T^{2} \)
41 \( 1 + 2.65e8T + 5.50e17T^{2} \)
43 \( 1 - 5.47e8T + 9.29e17T^{2} \)
47 \( 1 + 7.16e8T + 2.47e18T^{2} \)
53 \( 1 + 1.02e7T + 9.26e18T^{2} \)
61 \( 1 + 9.31e8T + 4.35e19T^{2} \)
67 \( 1 - 7.59e9T + 1.22e20T^{2} \)
71 \( 1 - 1.44e9T + 2.31e20T^{2} \)
73 \( 1 - 1.34e10T + 3.13e20T^{2} \)
79 \( 1 - 2.11e10T + 7.47e20T^{2} \)
83 \( 1 + 4.40e10T + 1.28e21T^{2} \)
89 \( 1 + 6.64e10T + 2.77e21T^{2} \)
97 \( 1 - 6.49e10T + 7.15e21T^{2} \)
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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

−10.18967104231007878300186099671, −9.479173539324568308524805598033, −8.940054695942241716131888147431, −7.84831321996150674090220040709, −6.79480741532873559834384397468, −5.67986361263560952365867268626, −3.83507383870243814732128577595, −3.34759410129019599501899156538, −1.53552561331692356965529278722, −0.67781972074853228613474334332, 0.67781972074853228613474334332, 1.53552561331692356965529278722, 3.34759410129019599501899156538, 3.83507383870243814732128577595, 5.67986361263560952365867268626, 6.79480741532873559834384397468, 7.84831321996150674090220040709, 8.940054695942241716131888147431, 9.479173539324568308524805598033, 10.18967104231007878300186099671

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