Properties

Label 2-177-1.1-c13-0-96
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  − 153.·2-s + 729·3-s + 1.52e4·4-s + 5.84e4·5-s − 1.11e5·6-s + 5.70e4·7-s − 1.08e6·8-s + 5.31e5·9-s − 8.95e6·10-s − 8.87e6·11-s + 1.11e7·12-s − 4.66e6·13-s − 8.73e6·14-s + 4.25e7·15-s + 4.12e7·16-s − 9.15e7·17-s − 8.14e7·18-s − 1.77e8·19-s + 8.92e8·20-s + 4.15e7·21-s + 1.35e9·22-s + 7.81e8·23-s − 7.91e8·24-s + 2.19e9·25-s + 7.14e8·26-s + 3.87e8·27-s + 8.71e8·28-s + ⋯
L(s)  = 1  − 1.69·2-s + 0.577·3-s + 1.86·4-s + 1.67·5-s − 0.977·6-s + 0.183·7-s − 1.46·8-s + 0.333·9-s − 2.83·10-s − 1.50·11-s + 1.07·12-s − 0.267·13-s − 0.310·14-s + 0.965·15-s + 0.614·16-s − 0.919·17-s − 0.564·18-s − 0.864·19-s + 3.11·20-s + 0.105·21-s + 2.55·22-s + 1.10·23-s − 0.845·24-s + 1.79·25-s + 0.453·26-s + 0.192·27-s + 0.341·28-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 \( 1 + 153.T + 8.19e3T^{2} \)
5 \( 1 - 5.84e4T + 1.22e9T^{2} \)
7 \( 1 - 5.70e4T + 9.68e10T^{2} \)
11 \( 1 + 8.87e6T + 3.45e13T^{2} \)
13 \( 1 + 4.66e6T + 3.02e14T^{2} \)
17 \( 1 + 9.15e7T + 9.90e15T^{2} \)
19 \( 1 + 1.77e8T + 4.20e16T^{2} \)
23 \( 1 - 7.81e8T + 5.04e17T^{2} \)
29 \( 1 + 3.99e9T + 1.02e19T^{2} \)
31 \( 1 - 5.30e9T + 2.44e19T^{2} \)
37 \( 1 - 1.42e10T + 2.43e20T^{2} \)
41 \( 1 - 3.50e10T + 9.25e20T^{2} \)
43 \( 1 + 2.93e10T + 1.71e21T^{2} \)
47 \( 1 + 4.10e10T + 5.46e21T^{2} \)
53 \( 1 - 2.99e11T + 2.60e22T^{2} \)
61 \( 1 + 6.54e11T + 1.61e23T^{2} \)
67 \( 1 - 1.22e12T + 5.48e23T^{2} \)
71 \( 1 + 1.52e12T + 1.16e24T^{2} \)
73 \( 1 - 1.65e12T + 1.67e24T^{2} \)
79 \( 1 + 2.59e12T + 4.66e24T^{2} \)
83 \( 1 + 2.52e12T + 8.87e24T^{2} \)
89 \( 1 - 6.85e12T + 2.19e25T^{2} \)
97 \( 1 + 5.66e12T + 6.73e25T^{2} \)
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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

−9.707496004924245265948882207485, −8.984667568142692525646496364674, −8.154065058148616411841200192239, −7.13222401460446291250088401856, −6.12038578559237782093496052695, −4.87118454414644309378069538169, −2.60781716758110792112805645529, −2.25934008696668939153179831636, −1.24330717681974243454040292261, 0, 1.24330717681974243454040292261, 2.25934008696668939153179831636, 2.60781716758110792112805645529, 4.87118454414644309378069538169, 6.12038578559237782093496052695, 7.13222401460446291250088401856, 8.154065058148616411841200192239, 8.984667568142692525646496364674, 9.707496004924245265948882207485

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