Properties

Label 2-177-1.1-c13-0-68
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $189.798$
Root an. cond. $13.7767$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 83.1·2-s − 729·3-s − 1.27e3·4-s − 2.69e4·5-s − 6.06e4·6-s − 1.94e5·7-s − 7.87e5·8-s + 5.31e5·9-s − 2.23e6·10-s + 3.56e4·11-s + 9.32e5·12-s + 2.52e7·13-s − 1.61e7·14-s + 1.96e7·15-s − 5.49e7·16-s + 9.48e7·17-s + 4.41e7·18-s − 2.94e8·19-s + 3.44e7·20-s + 1.41e8·21-s + 2.96e6·22-s − 6.22e8·23-s + 5.74e8·24-s − 4.96e8·25-s + 2.10e9·26-s − 3.87e8·27-s + 2.48e8·28-s + ⋯
L(s)  = 1  + 0.918·2-s − 0.577·3-s − 0.156·4-s − 0.770·5-s − 0.530·6-s − 0.625·7-s − 1.06·8-s + 0.333·9-s − 0.707·10-s + 0.00606·11-s + 0.0901·12-s + 1.45·13-s − 0.574·14-s + 0.444·15-s − 0.819·16-s + 0.953·17-s + 0.306·18-s − 1.43·19-s + 0.120·20-s + 0.360·21-s + 0.00557·22-s − 0.877·23-s + 0.613·24-s − 0.406·25-s + 1.33·26-s − 0.192·27-s + 0.0975·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(189.798\)
Root analytic conductor: \(13.7767\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 729T \)
59 \( 1 + 4.21e10T \)
good2 \( 1 - 83.1T + 8.19e3T^{2} \)
5 \( 1 + 2.69e4T + 1.22e9T^{2} \)
7 \( 1 + 1.94e5T + 9.68e10T^{2} \)
11 \( 1 - 3.56e4T + 3.45e13T^{2} \)
13 \( 1 - 2.52e7T + 3.02e14T^{2} \)
17 \( 1 - 9.48e7T + 9.90e15T^{2} \)
19 \( 1 + 2.94e8T + 4.20e16T^{2} \)
23 \( 1 + 6.22e8T + 5.04e17T^{2} \)
29 \( 1 - 6.22e8T + 1.02e19T^{2} \)
31 \( 1 - 7.97e9T + 2.44e19T^{2} \)
37 \( 1 - 2.21e10T + 2.43e20T^{2} \)
41 \( 1 - 1.90e10T + 9.25e20T^{2} \)
43 \( 1 - 7.56e10T + 1.71e21T^{2} \)
47 \( 1 - 8.66e10T + 5.46e21T^{2} \)
53 \( 1 - 1.36e10T + 2.60e22T^{2} \)
61 \( 1 + 3.35e11T + 1.61e23T^{2} \)
67 \( 1 + 9.33e11T + 5.48e23T^{2} \)
71 \( 1 + 1.39e11T + 1.16e24T^{2} \)
73 \( 1 - 1.56e11T + 1.67e24T^{2} \)
79 \( 1 + 2.07e12T + 4.66e24T^{2} \)
83 \( 1 + 1.77e12T + 8.87e24T^{2} \)
89 \( 1 + 7.85e12T + 2.19e25T^{2} \)
97 \( 1 - 5.22e11T + 6.73e25T^{2} \)
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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

−9.993068144134803518326113130970, −8.789158543090876951113562779992, −7.79276142409529848034324127543, −6.20926357332788765075740270183, −5.94093316814466142714324286927, −4.34877973747448181193372955238, −3.94688531624442835640754845874, −2.78595611963073345518946762032, −0.974554989461351816814433473100, 0, 0.974554989461351816814433473100, 2.78595611963073345518946762032, 3.94688531624442835640754845874, 4.34877973747448181193372955238, 5.94093316814466142714324286927, 6.20926357332788765075740270183, 7.79276142409529848034324127543, 8.789158543090876951113562779992, 9.993068144134803518326113130970

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