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  + 0.618·2-s − 3-s − 1.61·4-s − 2.23·5-s − 0.618·6-s − 2.38·7-s − 2.23·8-s + 9-s − 1.38·10-s + 2.23·11-s + 1.61·12-s − 6.23·13-s − 1.47·14-s + 2.23·15-s + 1.85·16-s + 1.85·17-s + 0.618·18-s + 3.09·19-s + 3.61·20-s + 2.38·21-s + 1.38·22-s − 4.61·23-s + 2.23·24-s − 3.85·26-s − 27-s + 3.85·28-s + 6.38·29-s + ⋯
L(s)  = 1  + 0.437·2-s − 0.577·3-s − 0.809·4-s − 0.999·5-s − 0.252·6-s − 0.900·7-s − 0.790·8-s + 0.333·9-s − 0.437·10-s + 0.674·11-s + 0.467·12-s − 1.72·13-s − 0.393·14-s + 0.577·15-s + 0.463·16-s + 0.449·17-s + 0.145·18-s + 0.708·19-s + 0.809·20-s + 0.519·21-s + 0.294·22-s − 0.962·23-s + 0.456·24-s − 0.755·26-s − 0.192·27-s + 0.728·28-s + 1.18·29-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 \( 1 - 0.618T + 2T^{2} \)
5 \( 1 + 2.23T + 5T^{2} \)
7 \( 1 + 2.38T + 7T^{2} \)
11 \( 1 - 2.23T + 11T^{2} \)
13 \( 1 + 6.23T + 13T^{2} \)
17 \( 1 - 1.85T + 17T^{2} \)
19 \( 1 - 3.09T + 19T^{2} \)
23 \( 1 + 4.61T + 23T^{2} \)
29 \( 1 - 6.38T + 29T^{2} \)
31 \( 1 + 10.5T + 31T^{2} \)
37 \( 1 + 0.145T + 37T^{2} \)
41 \( 1 - 8.09T + 41T^{2} \)
43 \( 1 + 8.70T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 - 6.23T + 53T^{2} \)
61 \( 1 + 3.14T + 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 + 7.94T + 71T^{2} \)
73 \( 1 - 0.854T + 73T^{2} \)
79 \( 1 + 3T + 79T^{2} \)
83 \( 1 + 1.61T + 83T^{2} \)
89 \( 1 + 13.7T + 89T^{2} \)
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

−12.22217381661985888528078665499, −11.65351909314581885259103508977, −10.02773240521845233365231295530, −9.414588112306327463522950750088, −7.966912615309547791588210516276, −6.87996061281009109379973992105, −5.54848578879047455243056290371, −4.42357676262472025458636139018, −3.36750047154044011266289456885, 0, 3.36750047154044011266289456885, 4.42357676262472025458636139018, 5.54848578879047455243056290371, 6.87996061281009109379973992105, 7.966912615309547791588210516276, 9.414588112306327463522950750088, 10.02773240521845233365231295530, 11.65351909314581885259103508977, 12.22217381661985888528078665499

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