Properties

Degree $2$
Conductor $177$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·2-s − 3-s + 0.618·4-s + 2.23·5-s + 1.61·6-s − 4.61·7-s + 2.23·8-s + 9-s − 3.61·10-s − 2.23·11-s − 0.618·12-s − 1.76·13-s + 7.47·14-s − 2.23·15-s − 4.85·16-s − 4.85·17-s − 1.61·18-s − 8.09·19-s + 1.38·20-s + 4.61·21-s + 3.61·22-s − 2.38·23-s − 2.23·24-s + 2.85·26-s − 27-s − 2.85·28-s + 8.61·29-s + ⋯
L(s)  = 1  − 1.14·2-s − 0.577·3-s + 0.309·4-s + 0.999·5-s + 0.660·6-s − 1.74·7-s + 0.790·8-s + 0.333·9-s − 1.14·10-s − 0.674·11-s − 0.178·12-s − 0.489·13-s + 1.99·14-s − 0.577·15-s − 1.21·16-s − 1.17·17-s − 0.381·18-s − 1.85·19-s + 0.309·20-s + 1.00·21-s + 0.771·22-s − 0.496·23-s − 0.456·24-s + 0.559·26-s − 0.192·27-s − 0.539·28-s + 1.60·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
59 \( 1 + T \)
good2 \( 1 + 1.61T + 2T^{2} \)
5 \( 1 - 2.23T + 5T^{2} \)
7 \( 1 + 4.61T + 7T^{2} \)
11 \( 1 + 2.23T + 11T^{2} \)
13 \( 1 + 1.76T + 13T^{2} \)
17 \( 1 + 4.85T + 17T^{2} \)
19 \( 1 + 8.09T + 19T^{2} \)
23 \( 1 + 2.38T + 23T^{2} \)
29 \( 1 - 8.61T + 29T^{2} \)
31 \( 1 - 9.56T + 31T^{2} \)
37 \( 1 + 6.85T + 37T^{2} \)
41 \( 1 + 3.09T + 41T^{2} \)
43 \( 1 - 4.70T + 43T^{2} \)
47 \( 1 + 4.14T + 47T^{2} \)
53 \( 1 - 1.76T + 53T^{2} \)
61 \( 1 + 9.85T + 61T^{2} \)
67 \( 1 + 2.70T + 67T^{2} \)
71 \( 1 - 9.94T + 71T^{2} \)
73 \( 1 + 5.85T + 73T^{2} \)
79 \( 1 + 3T + 79T^{2} \)
83 \( 1 - 0.618T + 83T^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 - 3T + 97T^{2} \)
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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

−12.25610894211952283070726136928, −10.52442417640008856256494621834, −10.22955949647475949358957612481, −9.371416471870876903956331230307, −8.390133296020820638503016036415, −6.78262173914783114446061021898, −6.20987270420816684246939244881, −4.57452023650093189296413504016, −2.38059346977599866708154952726, 0, 2.38059346977599866708154952726, 4.57452023650093189296413504016, 6.20987270420816684246939244881, 6.78262173914783114446061021898, 8.390133296020820638503016036415, 9.371416471870876903956331230307, 10.22955949647475949358957612481, 10.52442417640008856256494621834, 12.25610894211952283070726136928

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