Properties

Label 2-177-1.1-c9-0-29
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  + 10.2·2-s + 81·3-s − 406.·4-s + 1.02e3·5-s + 832.·6-s − 5.40e3·7-s − 9.44e3·8-s + 6.56e3·9-s + 1.04e4·10-s + 8.30e4·11-s − 3.29e4·12-s + 7.97e4·13-s − 5.56e4·14-s + 8.27e4·15-s + 1.10e5·16-s − 3.71e3·17-s + 6.74e4·18-s − 9.20e5·19-s − 4.14e5·20-s − 4.38e5·21-s + 8.54e5·22-s − 8.02e5·23-s − 7.64e5·24-s − 9.10e5·25-s + 8.20e5·26-s + 5.31e5·27-s + 2.19e6·28-s + ⋯
L(s)  = 1  + 0.454·2-s + 0.577·3-s − 0.793·4-s + 0.730·5-s + 0.262·6-s − 0.851·7-s − 0.814·8-s + 0.333·9-s + 0.332·10-s + 1.71·11-s − 0.458·12-s + 0.774·13-s − 0.386·14-s + 0.421·15-s + 0.423·16-s − 0.0107·17-s + 0.151·18-s − 1.62·19-s − 0.579·20-s − 0.491·21-s + 0.777·22-s − 0.597·23-s − 0.470·24-s − 0.466·25-s + 0.351·26-s + 0.192·27-s + 0.675·28-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

\(L(5)\) \(\approx\) \(3.080752012\)
\(L(\frac12)\) \(\approx\) \(3.080752012\)
\(L(\frac{11}{2})\) 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 \( 1 - 10.2T + 512T^{2} \)
5 \( 1 - 1.02e3T + 1.95e6T^{2} \)
7 \( 1 + 5.40e3T + 4.03e7T^{2} \)
11 \( 1 - 8.30e4T + 2.35e9T^{2} \)
13 \( 1 - 7.97e4T + 1.06e10T^{2} \)
17 \( 1 + 3.71e3T + 1.18e11T^{2} \)
19 \( 1 + 9.20e5T + 3.22e11T^{2} \)
23 \( 1 + 8.02e5T + 1.80e12T^{2} \)
29 \( 1 - 1.05e6T + 1.45e13T^{2} \)
31 \( 1 + 4.68e6T + 2.64e13T^{2} \)
37 \( 1 - 1.75e7T + 1.29e14T^{2} \)
41 \( 1 - 2.26e7T + 3.27e14T^{2} \)
43 \( 1 - 3.56e7T + 5.02e14T^{2} \)
47 \( 1 - 1.91e7T + 1.11e15T^{2} \)
53 \( 1 + 4.74e7T + 3.29e15T^{2} \)
61 \( 1 + 1.71e7T + 1.16e16T^{2} \)
67 \( 1 - 2.36e8T + 2.72e16T^{2} \)
71 \( 1 - 3.46e8T + 4.58e16T^{2} \)
73 \( 1 - 4.22e8T + 5.88e16T^{2} \)
79 \( 1 - 5.45e8T + 1.19e17T^{2} \)
83 \( 1 + 5.05e7T + 1.86e17T^{2} \)
89 \( 1 + 1.50e8T + 3.50e17T^{2} \)
97 \( 1 + 1.06e9T + 7.60e17T^{2} \)
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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

−10.96522833004734671264962915360, −9.468279137737702176331985126631, −9.363522707338759127014289853244, −8.212790310294632912889193446154, −6.46841494786507196147024333685, −5.96472502108117357349092437551, −4.22853995610923228981621627077, −3.66812094240484767641977490792, −2.20310548524213176557572020977, −0.804008435691693438503015685947, 0.804008435691693438503015685947, 2.20310548524213176557572020977, 3.66812094240484767641977490792, 4.22853995610923228981621627077, 5.96472502108117357349092437551, 6.46841494786507196147024333685, 8.212790310294632912889193446154, 9.363522707338759127014289853244, 9.468279137737702176331985126631, 10.96522833004734671264962915360

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