Properties

Label 2-177-1.1-c13-0-10
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  − 32.6·2-s − 729·3-s − 7.12e3·4-s − 4.93e4·5-s + 2.37e4·6-s − 5.93e5·7-s + 4.99e5·8-s + 5.31e5·9-s + 1.61e6·10-s − 7.46e6·11-s + 5.19e6·12-s − 3.21e7·13-s + 1.93e7·14-s + 3.59e7·15-s + 4.20e7·16-s − 1.18e7·17-s − 1.73e7·18-s − 3.23e8·19-s + 3.52e8·20-s + 4.32e8·21-s + 2.43e8·22-s − 7.23e8·23-s − 3.64e8·24-s + 1.21e9·25-s + 1.04e9·26-s − 3.87e8·27-s + 4.22e9·28-s + ⋯
L(s)  = 1  − 0.360·2-s − 0.577·3-s − 0.870·4-s − 1.41·5-s + 0.208·6-s − 1.90·7-s + 0.673·8-s + 0.333·9-s + 0.509·10-s − 1.27·11-s + 0.502·12-s − 1.84·13-s + 0.686·14-s + 0.816·15-s + 0.627·16-s − 0.119·17-s − 0.120·18-s − 1.57·19-s + 1.22·20-s + 1.10·21-s + 0.457·22-s − 1.01·23-s − 0.389·24-s + 0.997·25-s + 0.665·26-s − 0.192·27-s + 1.65·28-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 \( 1 + 32.6T + 8.19e3T^{2} \)
5 \( 1 + 4.93e4T + 1.22e9T^{2} \)
7 \( 1 + 5.93e5T + 9.68e10T^{2} \)
11 \( 1 + 7.46e6T + 3.45e13T^{2} \)
13 \( 1 + 3.21e7T + 3.02e14T^{2} \)
17 \( 1 + 1.18e7T + 9.90e15T^{2} \)
19 \( 1 + 3.23e8T + 4.20e16T^{2} \)
23 \( 1 + 7.23e8T + 5.04e17T^{2} \)
29 \( 1 + 1.75e9T + 1.02e19T^{2} \)
31 \( 1 + 3.99e9T + 2.44e19T^{2} \)
37 \( 1 - 1.32e7T + 2.43e20T^{2} \)
41 \( 1 + 2.57e10T + 9.25e20T^{2} \)
43 \( 1 - 5.80e10T + 1.71e21T^{2} \)
47 \( 1 + 6.92e10T + 5.46e21T^{2} \)
53 \( 1 - 3.07e11T + 2.60e22T^{2} \)
61 \( 1 - 1.31e11T + 1.61e23T^{2} \)
67 \( 1 + 2.68e11T + 5.48e23T^{2} \)
71 \( 1 + 1.88e12T + 1.16e24T^{2} \)
73 \( 1 + 7.87e11T + 1.67e24T^{2} \)
79 \( 1 + 1.97e12T + 4.66e24T^{2} \)
83 \( 1 + 2.48e11T + 8.87e24T^{2} \)
89 \( 1 + 5.76e11T + 2.19e25T^{2} \)
97 \( 1 - 6.37e12T + 6.73e25T^{2} \)
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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

−10.01467061618289374332554490553, −8.910826940334968784803514780173, −7.74279695606033976626879282901, −7.07040287386284599645018330262, −5.69251641420637026112796825400, −4.52120107747954841842359387045, −3.73877138961289353518056263492, −2.49550349787349474346430479494, −0.26144212581851694506302727379, 0, 0.26144212581851694506302727379, 2.49550349787349474346430479494, 3.73877138961289353518056263492, 4.52120107747954841842359387045, 5.69251641420637026112796825400, 7.07040287386284599645018330262, 7.74279695606033976626879282901, 8.910826940334968784803514780173, 10.01467061618289374332554490553

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