Properties

Label 2-177-1.1-c9-0-26
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  − 44.6·2-s − 81·3-s + 1.48e3·4-s − 714.·5-s + 3.61e3·6-s − 1.00e4·7-s − 4.33e4·8-s + 6.56e3·9-s + 3.19e4·10-s − 1.69e4·11-s − 1.20e5·12-s − 1.93e4·13-s + 4.50e5·14-s + 5.78e4·15-s + 1.17e6·16-s + 3.36e5·17-s − 2.93e5·18-s − 7.88e5·19-s − 1.05e6·20-s + 8.16e5·21-s + 7.55e5·22-s + 1.24e5·23-s + 3.51e6·24-s − 1.44e6·25-s + 8.62e5·26-s − 5.31e5·27-s − 1.49e7·28-s + ⋯
L(s)  = 1  − 1.97·2-s − 0.577·3-s + 2.89·4-s − 0.511·5-s + 1.13·6-s − 1.58·7-s − 3.74·8-s + 0.333·9-s + 1.00·10-s − 0.348·11-s − 1.67·12-s − 0.187·13-s + 3.13·14-s + 0.295·15-s + 4.48·16-s + 0.978·17-s − 0.657·18-s − 1.38·19-s − 1.47·20-s + 0.916·21-s + 0.687·22-s + 0.0924·23-s + 2.16·24-s − 0.738·25-s + 0.370·26-s − 0.192·27-s − 4.59·28-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 \( 1 + 44.6T + 512T^{2} \)
5 \( 1 + 714.T + 1.95e6T^{2} \)
7 \( 1 + 1.00e4T + 4.03e7T^{2} \)
11 \( 1 + 1.69e4T + 2.35e9T^{2} \)
13 \( 1 + 1.93e4T + 1.06e10T^{2} \)
17 \( 1 - 3.36e5T + 1.18e11T^{2} \)
19 \( 1 + 7.88e5T + 3.22e11T^{2} \)
23 \( 1 - 1.24e5T + 1.80e12T^{2} \)
29 \( 1 - 4.52e6T + 1.45e13T^{2} \)
31 \( 1 + 5.79e6T + 2.64e13T^{2} \)
37 \( 1 - 3.64e6T + 1.29e14T^{2} \)
41 \( 1 - 1.96e7T + 3.27e14T^{2} \)
43 \( 1 + 3.00e6T + 5.02e14T^{2} \)
47 \( 1 + 3.78e7T + 1.11e15T^{2} \)
53 \( 1 - 8.32e7T + 3.29e15T^{2} \)
61 \( 1 - 1.36e8T + 1.16e16T^{2} \)
67 \( 1 + 4.31e7T + 2.72e16T^{2} \)
71 \( 1 - 2.79e8T + 4.58e16T^{2} \)
73 \( 1 + 1.26e8T + 5.88e16T^{2} \)
79 \( 1 - 5.15e8T + 1.19e17T^{2} \)
83 \( 1 + 4.95e8T + 1.86e17T^{2} \)
89 \( 1 - 3.38e8T + 3.50e17T^{2} \)
97 \( 1 - 1.18e9T + 7.60e17T^{2} \)
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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

−10.22093558811263576365367135105, −9.684759376246685710824238226498, −8.580952454804470183538270872880, −7.55224580497059979775187575187, −6.67087762072626538161918478009, −5.87244582432605089826166923091, −3.51322416617707031377994988097, −2.31349123789614063390489146856, −0.75268128211626770271952768569, 0, 0.75268128211626770271952768569, 2.31349123789614063390489146856, 3.51322416617707031377994988097, 5.87244582432605089826166923091, 6.67087762072626538161918478009, 7.55224580497059979775187575187, 8.580952454804470183538270872880, 9.684759376246685710824238226498, 10.22093558811263576365367135105

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