Properties

Label 2-177-1.1-c9-0-28
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  − 26.8·2-s + 81·3-s + 209.·4-s − 2.74e3·5-s − 2.17e3·6-s − 8.87e3·7-s + 8.12e3·8-s + 6.56e3·9-s + 7.36e4·10-s − 6.17e4·11-s + 1.69e4·12-s − 9.34e4·13-s + 2.38e5·14-s − 2.22e5·15-s − 3.25e5·16-s + 5.40e5·17-s − 1.76e5·18-s + 5.91e5·19-s − 5.74e5·20-s − 7.19e5·21-s + 1.65e6·22-s + 5.71e5·23-s + 6.58e5·24-s + 5.56e6·25-s + 2.50e6·26-s + 5.31e5·27-s − 1.85e6·28-s + ⋯
L(s)  = 1  − 1.18·2-s + 0.577·3-s + 0.408·4-s − 1.96·5-s − 0.685·6-s − 1.39·7-s + 0.701·8-s + 0.333·9-s + 2.32·10-s − 1.27·11-s + 0.236·12-s − 0.907·13-s + 1.65·14-s − 1.13·15-s − 1.24·16-s + 1.56·17-s − 0.395·18-s + 1.04·19-s − 0.802·20-s − 0.806·21-s + 1.50·22-s + 0.425·23-s + 0.405·24-s + 2.85·25-s + 1.07·26-s + 0.192·27-s − 0.571·28-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 \( 1 + 26.8T + 512T^{2} \)
5 \( 1 + 2.74e3T + 1.95e6T^{2} \)
7 \( 1 + 8.87e3T + 4.03e7T^{2} \)
11 \( 1 + 6.17e4T + 2.35e9T^{2} \)
13 \( 1 + 9.34e4T + 1.06e10T^{2} \)
17 \( 1 - 5.40e5T + 1.18e11T^{2} \)
19 \( 1 - 5.91e5T + 3.22e11T^{2} \)
23 \( 1 - 5.71e5T + 1.80e12T^{2} \)
29 \( 1 + 1.20e6T + 1.45e13T^{2} \)
31 \( 1 + 3.72e6T + 2.64e13T^{2} \)
37 \( 1 - 1.40e7T + 1.29e14T^{2} \)
41 \( 1 + 7.71e6T + 3.27e14T^{2} \)
43 \( 1 - 3.91e6T + 5.02e14T^{2} \)
47 \( 1 + 5.70e7T + 1.11e15T^{2} \)
53 \( 1 - 4.11e7T + 3.29e15T^{2} \)
61 \( 1 - 5.31e7T + 1.16e16T^{2} \)
67 \( 1 - 2.25e8T + 2.72e16T^{2} \)
71 \( 1 + 2.29e8T + 4.58e16T^{2} \)
73 \( 1 - 1.11e8T + 5.88e16T^{2} \)
79 \( 1 - 7.39e7T + 1.19e17T^{2} \)
83 \( 1 - 1.07e8T + 1.86e17T^{2} \)
89 \( 1 + 6.13e8T + 3.50e17T^{2} \)
97 \( 1 - 6.55e8T + 7.60e17T^{2} \)
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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

−10.13960461666925492579742903707, −9.545555902510325292186301380105, −8.336735346139023023819145185067, −7.61373841732540258849398634977, −7.17508360670281534929466252785, −5.02035697316529452829356784736, −3.62516427876987312187203470181, −2.85743321598984424921079458211, −0.78487227138101284743329803731, 0, 0.78487227138101284743329803731, 2.85743321598984424921079458211, 3.62516427876987312187203470181, 5.02035697316529452829356784736, 7.17508360670281534929466252785, 7.61373841732540258849398634977, 8.336735346139023023819145185067, 9.545555902510325292186301380105, 10.13960461666925492579742903707

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