Properties

Label 2-177-1.1-c9-0-7
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 34.5·2-s − 81·3-s + 678.·4-s − 2.12e3·5-s − 2.79e3·6-s − 1.10e4·7-s + 5.73e3·8-s + 6.56e3·9-s − 7.33e4·10-s − 1.28e4·11-s − 5.49e4·12-s − 9.83e4·13-s − 3.82e5·14-s + 1.72e5·15-s − 1.49e5·16-s − 5.52e4·17-s + 2.26e5·18-s + 1.34e5·19-s − 1.44e6·20-s + 8.97e5·21-s − 4.43e5·22-s + 1.76e6·23-s − 4.64e5·24-s + 2.56e6·25-s − 3.39e6·26-s − 5.31e5·27-s − 7.51e6·28-s + ⋯
L(s)  = 1  + 1.52·2-s − 0.577·3-s + 1.32·4-s − 1.52·5-s − 0.880·6-s − 1.74·7-s + 0.495·8-s + 0.333·9-s − 2.31·10-s − 0.264·11-s − 0.764·12-s − 0.955·13-s − 2.66·14-s + 0.878·15-s − 0.569·16-s − 0.160·17-s + 0.508·18-s + 0.237·19-s − 2.01·20-s + 1.00·21-s − 0.403·22-s + 1.31·23-s − 0.285·24-s + 1.31·25-s − 1.45·26-s − 0.192·27-s − 2.31·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.9388169028\)
\(L(\frac12)\) \(\approx\) \(0.9388169028\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 81T \)
59 \( 1 - 1.21e7T \)
good2 \( 1 - 34.5T + 512T^{2} \)
5 \( 1 + 2.12e3T + 1.95e6T^{2} \)
7 \( 1 + 1.10e4T + 4.03e7T^{2} \)
11 \( 1 + 1.28e4T + 2.35e9T^{2} \)
13 \( 1 + 9.83e4T + 1.06e10T^{2} \)
17 \( 1 + 5.52e4T + 1.18e11T^{2} \)
19 \( 1 - 1.34e5T + 3.22e11T^{2} \)
23 \( 1 - 1.76e6T + 1.80e12T^{2} \)
29 \( 1 + 1.27e6T + 1.45e13T^{2} \)
31 \( 1 - 6.55e5T + 2.64e13T^{2} \)
37 \( 1 + 1.12e7T + 1.29e14T^{2} \)
41 \( 1 - 9.02e6T + 3.27e14T^{2} \)
43 \( 1 + 3.12e7T + 5.02e14T^{2} \)
47 \( 1 - 4.08e7T + 1.11e15T^{2} \)
53 \( 1 - 5.63e7T + 3.29e15T^{2} \)
61 \( 1 + 5.05e7T + 1.16e16T^{2} \)
67 \( 1 + 2.08e8T + 2.72e16T^{2} \)
71 \( 1 + 2.83e8T + 4.58e16T^{2} \)
73 \( 1 - 3.02e8T + 5.88e16T^{2} \)
79 \( 1 + 6.85e6T + 1.19e17T^{2} \)
83 \( 1 + 7.30e7T + 1.86e17T^{2} \)
89 \( 1 + 9.42e8T + 3.50e17T^{2} \)
97 \( 1 - 5.45e8T + 7.60e17T^{2} \)
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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

−11.50356430083815296393257159101, −10.37898279758439147679465438019, −9.061139593604756211835927577864, −7.31516047979433028328918022303, −6.76155873166727362092848725248, −5.54276062124378279630341976296, −4.51158270254807181681672150337, −3.55825495376209152323998613044, −2.81476950633213188528617792915, −0.36443002031314907851036081476, 0.36443002031314907851036081476, 2.81476950633213188528617792915, 3.55825495376209152323998613044, 4.51158270254807181681672150337, 5.54276062124378279630341976296, 6.76155873166727362092848725248, 7.31516047979433028328918022303, 9.061139593604756211835927577864, 10.37898279758439147679465438019, 11.50356430083815296393257159101

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