Properties

Label 2-177-1.1-c9-0-60
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  + 2.64·2-s − 81·3-s − 504.·4-s + 1.26e3·5-s − 214.·6-s + 7.40e3·7-s − 2.69e3·8-s + 6.56e3·9-s + 3.35e3·10-s − 8.39e4·11-s + 4.09e4·12-s − 2.09e4·13-s + 1.95e4·14-s − 1.02e5·15-s + 2.51e5·16-s + 2.36e5·17-s + 1.73e4·18-s − 3.82e5·19-s − 6.40e5·20-s − 5.99e5·21-s − 2.22e5·22-s + 6.05e5·23-s + 2.18e5·24-s − 3.45e5·25-s − 5.55e4·26-s − 5.31e5·27-s − 3.73e6·28-s + ⋯
L(s)  = 1  + 0.116·2-s − 0.577·3-s − 0.986·4-s + 0.907·5-s − 0.0675·6-s + 1.16·7-s − 0.232·8-s + 0.333·9-s + 0.106·10-s − 1.72·11-s + 0.569·12-s − 0.203·13-s + 0.136·14-s − 0.523·15-s + 0.959·16-s + 0.686·17-s + 0.0389·18-s − 0.674·19-s − 0.894·20-s − 0.672·21-s − 0.202·22-s + 0.451·23-s + 0.134·24-s − 0.176·25-s − 0.0238·26-s − 0.192·27-s − 1.14·28-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 \( 1 - 2.64T + 512T^{2} \)
5 \( 1 - 1.26e3T + 1.95e6T^{2} \)
7 \( 1 - 7.40e3T + 4.03e7T^{2} \)
11 \( 1 + 8.39e4T + 2.35e9T^{2} \)
13 \( 1 + 2.09e4T + 1.06e10T^{2} \)
17 \( 1 - 2.36e5T + 1.18e11T^{2} \)
19 \( 1 + 3.82e5T + 3.22e11T^{2} \)
23 \( 1 - 6.05e5T + 1.80e12T^{2} \)
29 \( 1 - 1.75e6T + 1.45e13T^{2} \)
31 \( 1 + 2.07e6T + 2.64e13T^{2} \)
37 \( 1 - 1.60e7T + 1.29e14T^{2} \)
41 \( 1 - 3.00e7T + 3.27e14T^{2} \)
43 \( 1 + 2.29e6T + 5.02e14T^{2} \)
47 \( 1 + 6.22e7T + 1.11e15T^{2} \)
53 \( 1 - 5.08e7T + 3.29e15T^{2} \)
61 \( 1 - 1.27e8T + 1.16e16T^{2} \)
67 \( 1 - 1.93e7T + 2.72e16T^{2} \)
71 \( 1 + 2.57e8T + 4.58e16T^{2} \)
73 \( 1 + 4.11e8T + 5.88e16T^{2} \)
79 \( 1 - 3.74e7T + 1.19e17T^{2} \)
83 \( 1 + 4.85e8T + 1.86e17T^{2} \)
89 \( 1 + 1.10e9T + 3.50e17T^{2} \)
97 \( 1 + 1.73e8T + 7.60e17T^{2} \)
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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

−10.41388153441048489403035467412, −9.719307888087306948871316454163, −8.419789398783560424407450917660, −7.61142583669770395076068895348, −5.85995200448052717066638107734, −5.22693841104513087816124130914, −4.41305310456153425889023621100, −2.60256808028057895457056078886, −1.27295304458092961401132801089, 0, 1.27295304458092961401132801089, 2.60256808028057895457056078886, 4.41305310456153425889023621100, 5.22693841104513087816124130914, 5.85995200448052717066638107734, 7.61142583669770395076068895348, 8.419789398783560424407450917660, 9.719307888087306948871316454163, 10.41388153441048489403035467412

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