Properties

Label 2-177-1.1-c9-0-3
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  − 17.2·2-s − 81·3-s − 215.·4-s + 470.·5-s + 1.39e3·6-s − 5.41e3·7-s + 1.25e4·8-s + 6.56e3·9-s − 8.10e3·10-s − 6.63e4·11-s + 1.74e4·12-s − 1.54e5·13-s + 9.31e4·14-s − 3.81e4·15-s − 1.05e5·16-s + 4.78e5·17-s − 1.12e5·18-s + 2.50e4·19-s − 1.01e5·20-s + 4.38e5·21-s + 1.14e6·22-s + 2.50e5·23-s − 1.01e6·24-s − 1.73e6·25-s + 2.66e6·26-s − 5.31e5·27-s + 1.16e6·28-s + ⋯
L(s)  = 1  − 0.760·2-s − 0.577·3-s − 0.421·4-s + 0.336·5-s + 0.439·6-s − 0.852·7-s + 1.08·8-s + 0.333·9-s − 0.256·10-s − 1.36·11-s + 0.243·12-s − 1.50·13-s + 0.648·14-s − 0.194·15-s − 0.401·16-s + 1.38·17-s − 0.253·18-s + 0.0440·19-s − 0.141·20-s + 0.491·21-s + 1.04·22-s + 0.186·23-s − 0.624·24-s − 0.886·25-s + 1.14·26-s − 0.192·27-s + 0.358·28-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

\(L(5)\) \(\approx\) \(0.1050969103\)
\(L(\frac12)\) \(\approx\) \(0.1050969103\)
\(L(\frac{11}{2})\) 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 \( 1 + 17.2T + 512T^{2} \)
5 \( 1 - 470.T + 1.95e6T^{2} \)
7 \( 1 + 5.41e3T + 4.03e7T^{2} \)
11 \( 1 + 6.63e4T + 2.35e9T^{2} \)
13 \( 1 + 1.54e5T + 1.06e10T^{2} \)
17 \( 1 - 4.78e5T + 1.18e11T^{2} \)
19 \( 1 - 2.50e4T + 3.22e11T^{2} \)
23 \( 1 - 2.50e5T + 1.80e12T^{2} \)
29 \( 1 - 2.66e5T + 1.45e13T^{2} \)
31 \( 1 + 7.24e6T + 2.64e13T^{2} \)
37 \( 1 + 1.81e7T + 1.29e14T^{2} \)
41 \( 1 + 1.23e7T + 3.27e14T^{2} \)
43 \( 1 + 1.58e7T + 5.02e14T^{2} \)
47 \( 1 + 3.87e6T + 1.11e15T^{2} \)
53 \( 1 + 2.96e7T + 3.29e15T^{2} \)
61 \( 1 + 1.03e8T + 1.16e16T^{2} \)
67 \( 1 + 2.36e8T + 2.72e16T^{2} \)
71 \( 1 + 8.61e7T + 4.58e16T^{2} \)
73 \( 1 + 2.99e8T + 5.88e16T^{2} \)
79 \( 1 - 2.81e7T + 1.19e17T^{2} \)
83 \( 1 - 6.97e8T + 1.86e17T^{2} \)
89 \( 1 - 1.72e8T + 3.50e17T^{2} \)
97 \( 1 + 5.76e8T + 7.60e17T^{2} \)
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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

−10.49053166031987364197686031163, −10.05427330853200713125052269957, −9.290461195503873986577944404870, −7.88694752669060606269224983002, −7.16904924362967532118199302879, −5.61614519307517951385194819413, −4.90353621712583699146975747362, −3.26130352366531466833856116566, −1.75848335638355450904595992862, −0.18086211324099767883527556936, 0.18086211324099767883527556936, 1.75848335638355450904595992862, 3.26130352366531466833856116566, 4.90353621712583699146975747362, 5.61614519307517951385194819413, 7.16904924362967532118199302879, 7.88694752669060606269224983002, 9.290461195503873986577944404870, 10.05427330853200713125052269957, 10.49053166031987364197686031163

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