Properties

Label 2-177-1.1-c3-0-13
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  + 2.04·2-s + 3·3-s − 3.80·4-s + 16.1·5-s + 6.14·6-s − 1.13·7-s − 24.1·8-s + 9·9-s + 33.1·10-s + 36.3·11-s − 11.4·12-s + 78.1·13-s − 2.33·14-s + 48.5·15-s − 19.1·16-s − 43.6·17-s + 18.4·18-s − 18.4·19-s − 61.5·20-s − 3.41·21-s + 74.4·22-s − 4.45·23-s − 72.5·24-s + 136.·25-s + 160.·26-s + 27·27-s + 4.33·28-s + ⋯
L(s)  = 1  + 0.724·2-s + 0.577·3-s − 0.475·4-s + 1.44·5-s + 0.418·6-s − 0.0615·7-s − 1.06·8-s + 0.333·9-s + 1.04·10-s + 0.995·11-s − 0.274·12-s + 1.66·13-s − 0.0445·14-s + 0.835·15-s − 0.299·16-s − 0.623·17-s + 0.241·18-s − 0.222·19-s − 0.687·20-s − 0.0355·21-s + 0.721·22-s − 0.0403·23-s − 0.617·24-s + 1.09·25-s + 1.20·26-s + 0.192·27-s + 0.0292·28-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

\(L(2)\) \(\approx\) \(3.274116022\)
\(L(\frac12)\) \(\approx\) \(3.274116022\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.04T + 8T^{2} \)
5 \( 1 - 16.1T + 125T^{2} \)
7 \( 1 + 1.13T + 343T^{2} \)
11 \( 1 - 36.3T + 1.33e3T^{2} \)
13 \( 1 - 78.1T + 2.19e3T^{2} \)
17 \( 1 + 43.6T + 4.91e3T^{2} \)
19 \( 1 + 18.4T + 6.85e3T^{2} \)
23 \( 1 + 4.45T + 1.21e4T^{2} \)
29 \( 1 + 161.T + 2.43e4T^{2} \)
31 \( 1 - 245.T + 2.97e4T^{2} \)
37 \( 1 + 173.T + 5.06e4T^{2} \)
41 \( 1 - 128.T + 6.89e4T^{2} \)
43 \( 1 + 229.T + 7.95e4T^{2} \)
47 \( 1 + 138.T + 1.03e5T^{2} \)
53 \( 1 + 168.T + 1.48e5T^{2} \)
61 \( 1 + 878.T + 2.26e5T^{2} \)
67 \( 1 - 361.T + 3.00e5T^{2} \)
71 \( 1 + 761.T + 3.57e5T^{2} \)
73 \( 1 + 404.T + 3.89e5T^{2} \)
79 \( 1 + 493.T + 4.93e5T^{2} \)
83 \( 1 - 398.T + 5.71e5T^{2} \)
89 \( 1 + 1.32e3T + 7.04e5T^{2} \)
97 \( 1 + 1.46e3T + 9.12e5T^{2} \)
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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

−12.69361092024778055243033353567, −11.34430483539179120365567535044, −10.01509007829104527896177156308, −9.147327395913674794090075787854, −8.521572267683515180099923037091, −6.51766502691944559645801961406, −5.87161961057290297224819379821, −4.44347615582221479879858583209, −3.25581763744218795927894628597, −1.59192764700765547128510977036, 1.59192764700765547128510977036, 3.25581763744218795927894628597, 4.44347615582221479879858583209, 5.87161961057290297224819379821, 6.51766502691944559645801961406, 8.521572267683515180099923037091, 9.147327395913674794090075787854, 10.01509007829104527896177156308, 11.34430483539179120365567535044, 12.69361092024778055243033353567

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