Properties

Label 2-177-1.1-c9-0-81
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 42.7·2-s − 81·3-s + 1.31e3·4-s − 657.·5-s − 3.46e3·6-s + 2.35e3·7-s + 3.45e4·8-s + 6.56e3·9-s − 2.81e4·10-s − 4.86e4·11-s − 1.06e5·12-s − 1.81e5·13-s + 1.00e5·14-s + 5.32e4·15-s + 8.01e5·16-s − 3.32e4·17-s + 2.80e5·18-s − 5.33e5·19-s − 8.66e5·20-s − 1.91e5·21-s − 2.07e6·22-s + 7.11e5·23-s − 2.79e6·24-s − 1.52e6·25-s − 7.76e6·26-s − 5.31e5·27-s + 3.11e6·28-s + ⋯
L(s)  = 1  + 1.89·2-s − 0.577·3-s + 2.57·4-s − 0.470·5-s − 1.09·6-s + 0.371·7-s + 2.97·8-s + 0.333·9-s − 0.889·10-s − 1.00·11-s − 1.48·12-s − 1.76·13-s + 0.702·14-s + 0.271·15-s + 3.05·16-s − 0.0964·17-s + 0.630·18-s − 0.938·19-s − 1.21·20-s − 0.214·21-s − 1.89·22-s + 0.530·23-s − 1.72·24-s − 0.778·25-s − 3.33·26-s − 0.192·27-s + 0.956·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 - 42.7T + 512T^{2} \)
5 \( 1 + 657.T + 1.95e6T^{2} \)
7 \( 1 - 2.35e3T + 4.03e7T^{2} \)
11 \( 1 + 4.86e4T + 2.35e9T^{2} \)
13 \( 1 + 1.81e5T + 1.06e10T^{2} \)
17 \( 1 + 3.32e4T + 1.18e11T^{2} \)
19 \( 1 + 5.33e5T + 3.22e11T^{2} \)
23 \( 1 - 7.11e5T + 1.80e12T^{2} \)
29 \( 1 - 3.51e5T + 1.45e13T^{2} \)
31 \( 1 + 1.44e6T + 2.64e13T^{2} \)
37 \( 1 - 1.69e7T + 1.29e14T^{2} \)
41 \( 1 + 2.49e7T + 3.27e14T^{2} \)
43 \( 1 - 1.83e7T + 5.02e14T^{2} \)
47 \( 1 + 1.25e7T + 1.11e15T^{2} \)
53 \( 1 + 7.15e7T + 3.29e15T^{2} \)
61 \( 1 + 4.58e7T + 1.16e16T^{2} \)
67 \( 1 - 1.49e8T + 2.72e16T^{2} \)
71 \( 1 + 3.47e8T + 4.58e16T^{2} \)
73 \( 1 + 8.25e7T + 5.88e16T^{2} \)
79 \( 1 + 5.99e8T + 1.19e17T^{2} \)
83 \( 1 + 2.20e7T + 1.86e17T^{2} \)
89 \( 1 - 1.41e8T + 3.50e17T^{2} \)
97 \( 1 - 5.95e8T + 7.60e17T^{2} \)
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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

−11.03894973010201049210232922104, −10.03779374546034963464758910717, −7.86917918982635585773730539953, −7.11078121836612213203421445337, −5.94915842671947591798585450370, −4.93166677621133122473454908414, −4.41490971711528722905749506584, −2.97077992643536654601707615557, −1.96424104645619242016026350928, 0, 1.96424104645619242016026350928, 2.97077992643536654601707615557, 4.41490971711528722905749506584, 4.93166677621133122473454908414, 5.94915842671947591798585450370, 7.11078121836612213203421445337, 7.86917918982635585773730539953, 10.03779374546034963464758910717, 11.03894973010201049210232922104

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