Properties

Label 2-177-1.1-c3-0-23
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.15·2-s + 3·3-s + 18.5·4-s + 1.69·5-s + 15.4·6-s − 11.0·7-s + 54.3·8-s + 9·9-s + 8.71·10-s − 10.1·11-s + 55.6·12-s + 29.5·13-s − 57.1·14-s + 5.07·15-s + 131.·16-s − 29.3·17-s + 46.3·18-s − 46.9·19-s + 31.3·20-s − 33.2·21-s − 52.2·22-s + 22.2·23-s + 163.·24-s − 122.·25-s + 152.·26-s + 27·27-s − 205.·28-s + ⋯
L(s)  = 1  + 1.82·2-s + 0.577·3-s + 2.31·4-s + 0.151·5-s + 1.05·6-s − 0.599·7-s + 2.40·8-s + 0.333·9-s + 0.275·10-s − 0.277·11-s + 1.33·12-s + 0.630·13-s − 1.09·14-s + 0.0873·15-s + 2.05·16-s − 0.419·17-s + 0.607·18-s − 0.566·19-s + 0.350·20-s − 0.345·21-s − 0.506·22-s + 0.201·23-s + 1.38·24-s − 0.977·25-s + 1.14·26-s + 0.192·27-s − 1.38·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.579728780\)
\(L(\frac12)\) \(\approx\) \(5.579728780\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
59 \( 1 - 59T \)
good2 \( 1 - 5.15T + 8T^{2} \)
5 \( 1 - 1.69T + 125T^{2} \)
7 \( 1 + 11.0T + 343T^{2} \)
11 \( 1 + 10.1T + 1.33e3T^{2} \)
13 \( 1 - 29.5T + 2.19e3T^{2} \)
17 \( 1 + 29.3T + 4.91e3T^{2} \)
19 \( 1 + 46.9T + 6.85e3T^{2} \)
23 \( 1 - 22.2T + 1.21e4T^{2} \)
29 \( 1 + 103.T + 2.43e4T^{2} \)
31 \( 1 + 52.5T + 2.97e4T^{2} \)
37 \( 1 - 5.51T + 5.06e4T^{2} \)
41 \( 1 - 201.T + 6.89e4T^{2} \)
43 \( 1 - 479.T + 7.95e4T^{2} \)
47 \( 1 + 133.T + 1.03e5T^{2} \)
53 \( 1 + 484.T + 1.48e5T^{2} \)
61 \( 1 - 578.T + 2.26e5T^{2} \)
67 \( 1 - 52.3T + 3.00e5T^{2} \)
71 \( 1 - 399.T + 3.57e5T^{2} \)
73 \( 1 + 1.04e3T + 3.89e5T^{2} \)
79 \( 1 - 269.T + 4.93e5T^{2} \)
83 \( 1 - 174.T + 5.71e5T^{2} \)
89 \( 1 - 1.38e3T + 7.04e5T^{2} \)
97 \( 1 - 628.T + 9.12e5T^{2} \)
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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

−12.79459258131240119002766808175, −11.52216483741965903971549933484, −10.61069309396721413978549342556, −9.265800223108257562482672059304, −7.76516664233551999785418742273, −6.59413564132328852160814775379, −5.71232996827151363675249787305, −4.33758419198079730335547234243, −3.36664135883380148032726680006, −2.14835795249272276993995709581, 2.14835795249272276993995709581, 3.36664135883380148032726680006, 4.33758419198079730335547234243, 5.71232996827151363675249787305, 6.59413564132328852160814775379, 7.76516664233551999785418742273, 9.265800223108257562482672059304, 10.61069309396721413978549342556, 11.52216483741965903971549933484, 12.79459258131240119002766808175

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