Properties

Label 2-177-1.1-c11-0-17
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 33.1·2-s + 243·3-s − 950.·4-s − 7.61e3·5-s − 8.05e3·6-s + 3.18e4·7-s + 9.93e4·8-s + 5.90e4·9-s + 2.52e5·10-s − 9.91e4·11-s − 2.30e5·12-s − 2.33e6·13-s − 1.05e6·14-s − 1.85e6·15-s − 1.34e6·16-s + 6.72e6·17-s − 1.95e6·18-s + 1.10e7·19-s + 7.23e6·20-s + 7.74e6·21-s + 3.28e6·22-s − 5.57e7·23-s + 2.41e7·24-s + 9.16e6·25-s + 7.74e7·26-s + 1.43e7·27-s − 3.02e7·28-s + ⋯
L(s)  = 1  − 0.732·2-s + 0.577·3-s − 0.463·4-s − 1.08·5-s − 0.422·6-s + 0.716·7-s + 1.07·8-s + 0.333·9-s + 0.797·10-s − 0.185·11-s − 0.267·12-s − 1.74·13-s − 0.524·14-s − 0.629·15-s − 0.320·16-s + 1.14·17-s − 0.244·18-s + 1.02·19-s + 0.505·20-s + 0.413·21-s + 0.135·22-s − 1.80·23-s + 0.618·24-s + 0.187·25-s + 1.27·26-s + 0.192·27-s − 0.332·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(0.8935225638\)
\(L(\frac12)\) \(\approx\) \(0.8935225638\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 243T \)
59 \( 1 - 7.14e8T \)
good2 \( 1 + 33.1T + 2.04e3T^{2} \)
5 \( 1 + 7.61e3T + 4.88e7T^{2} \)
7 \( 1 - 3.18e4T + 1.97e9T^{2} \)
11 \( 1 + 9.91e4T + 2.85e11T^{2} \)
13 \( 1 + 2.33e6T + 1.79e12T^{2} \)
17 \( 1 - 6.72e6T + 3.42e13T^{2} \)
19 \( 1 - 1.10e7T + 1.16e14T^{2} \)
23 \( 1 + 5.57e7T + 9.52e14T^{2} \)
29 \( 1 - 1.29e8T + 1.22e16T^{2} \)
31 \( 1 - 2.79e8T + 2.54e16T^{2} \)
37 \( 1 + 4.52e8T + 1.77e17T^{2} \)
41 \( 1 + 1.17e9T + 5.50e17T^{2} \)
43 \( 1 + 2.54e8T + 9.29e17T^{2} \)
47 \( 1 + 5.43e8T + 2.47e18T^{2} \)
53 \( 1 + 1.96e8T + 9.26e18T^{2} \)
61 \( 1 - 2.08e8T + 4.35e19T^{2} \)
67 \( 1 + 1.67e10T + 1.22e20T^{2} \)
71 \( 1 - 1.37e10T + 2.31e20T^{2} \)
73 \( 1 - 1.74e10T + 3.13e20T^{2} \)
79 \( 1 + 4.30e10T + 7.47e20T^{2} \)
83 \( 1 - 1.24e10T + 1.28e21T^{2} \)
89 \( 1 + 7.87e10T + 2.77e21T^{2} \)
97 \( 1 - 4.05e9T + 7.15e21T^{2} \)
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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

−10.15096309395775677874889616283, −9.823784581787111724705513837534, −8.277245768745755986328118232908, −8.017978139273954610174547000207, −7.18996498351941441917821545806, −5.11428414167419608831773902649, −4.35039100566052827458709528781, −3.13060753669128042880568287895, −1.68802234043928666228115252268, −0.47688717064910528209186432035, 0.47688717064910528209186432035, 1.68802234043928666228115252268, 3.13060753669128042880568287895, 4.35039100566052827458709528781, 5.11428414167419608831773902649, 7.18996498351941441917821545806, 8.017978139273954610174547000207, 8.277245768745755986328118232908, 9.823784581787111724705513837534, 10.15096309395775677874889616283

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