Properties

Label 2-177-1.1-c9-0-24
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  + 1.33·2-s − 81·3-s − 510.·4-s + 2.09e3·5-s − 108.·6-s + 107.·7-s − 1.36e3·8-s + 6.56e3·9-s + 2.79e3·10-s − 839.·11-s + 4.13e4·12-s + 3.40e4·13-s + 143.·14-s − 1.69e5·15-s + 2.59e5·16-s − 8.47e4·17-s + 8.76e3·18-s + 9.28e5·19-s − 1.06e6·20-s − 8.68e3·21-s − 1.12e3·22-s − 9.71e5·23-s + 1.10e5·24-s + 2.41e6·25-s + 4.54e4·26-s − 5.31e5·27-s − 5.46e4·28-s + ⋯
L(s)  = 1  + 0.0590·2-s − 0.577·3-s − 0.996·4-s + 1.49·5-s − 0.0340·6-s + 0.0168·7-s − 0.117·8-s + 0.333·9-s + 0.0883·10-s − 0.0172·11-s + 0.575·12-s + 0.330·13-s + 0.000996·14-s − 0.863·15-s + 0.989·16-s − 0.246·17-s + 0.0196·18-s + 1.63·19-s − 1.49·20-s − 0.00973·21-s − 0.00102·22-s − 0.724·23-s + 0.0680·24-s + 1.23·25-s + 0.0195·26-s − 0.192·27-s − 0.0168·28-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

\(L(5)\) \(\approx\) \(1.922173662\)
\(L(\frac12)\) \(\approx\) \(1.922173662\)
\(L(\frac{11}{2})\) 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 \( 1 - 1.33T + 512T^{2} \)
5 \( 1 - 2.09e3T + 1.95e6T^{2} \)
7 \( 1 - 107.T + 4.03e7T^{2} \)
11 \( 1 + 839.T + 2.35e9T^{2} \)
13 \( 1 - 3.40e4T + 1.06e10T^{2} \)
17 \( 1 + 8.47e4T + 1.18e11T^{2} \)
19 \( 1 - 9.28e5T + 3.22e11T^{2} \)
23 \( 1 + 9.71e5T + 1.80e12T^{2} \)
29 \( 1 - 5.34e5T + 1.45e13T^{2} \)
31 \( 1 + 1.99e6T + 2.64e13T^{2} \)
37 \( 1 + 9.60e6T + 1.29e14T^{2} \)
41 \( 1 - 1.18e7T + 3.27e14T^{2} \)
43 \( 1 + 1.85e7T + 5.02e14T^{2} \)
47 \( 1 + 1.01e7T + 1.11e15T^{2} \)
53 \( 1 + 3.57e7T + 3.29e15T^{2} \)
61 \( 1 + 6.03e7T + 1.16e16T^{2} \)
67 \( 1 - 2.83e8T + 2.72e16T^{2} \)
71 \( 1 - 3.57e8T + 4.58e16T^{2} \)
73 \( 1 - 3.74e8T + 5.88e16T^{2} \)
79 \( 1 - 6.00e7T + 1.19e17T^{2} \)
83 \( 1 - 6.16e8T + 1.86e17T^{2} \)
89 \( 1 - 1.58e8T + 3.50e17T^{2} \)
97 \( 1 - 1.10e9T + 7.60e17T^{2} \)
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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

−10.85075274353101980426840779772, −9.778600509661959303507539642536, −9.392625635194499930440013086177, −8.075742564046768143576225432036, −6.57080046881243057040588183662, −5.58258930168073430595451918747, −4.95024859552373340287748391903, −3.46650583855486516774469102380, −1.83207216206507510058882247409, −0.72863430862231156100194204877, 0.72863430862231156100194204877, 1.83207216206507510058882247409, 3.46650583855486516774469102380, 4.95024859552373340287748391903, 5.58258930168073430595451918747, 6.57080046881243057040588183662, 8.075742564046768143576225432036, 9.392625635194499930440013086177, 9.778600509661959303507539642536, 10.85075274353101980426840779772

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