Properties

Label 2-177-1.1-c3-0-10
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.07·2-s − 3·3-s + 17.7·4-s + 5.77·5-s + 15.2·6-s − 31.1·7-s − 49.2·8-s + 9·9-s − 29.2·10-s + 52.2·11-s − 53.1·12-s − 16.2·13-s + 157.·14-s − 17.3·15-s + 108.·16-s + 102.·17-s − 45.6·18-s + 46.2·19-s + 102.·20-s + 93.3·21-s − 264.·22-s − 99.1·23-s + 147.·24-s − 91.6·25-s + 82.3·26-s − 27·27-s − 551.·28-s + ⋯
L(s)  = 1  − 1.79·2-s − 0.577·3-s + 2.21·4-s + 0.516·5-s + 1.03·6-s − 1.68·7-s − 2.17·8-s + 0.333·9-s − 0.925·10-s + 1.43·11-s − 1.27·12-s − 0.346·13-s + 3.01·14-s − 0.298·15-s + 1.68·16-s + 1.46·17-s − 0.597·18-s + 0.558·19-s + 1.14·20-s + 0.970·21-s − 2.56·22-s − 0.898·23-s + 1.25·24-s − 0.733·25-s + 0.621·26-s − 0.192·27-s − 3.72·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
59 \( 1 - 59T \)
good2 \( 1 + 5.07T + 8T^{2} \)
5 \( 1 - 5.77T + 125T^{2} \)
7 \( 1 + 31.1T + 343T^{2} \)
11 \( 1 - 52.2T + 1.33e3T^{2} \)
13 \( 1 + 16.2T + 2.19e3T^{2} \)
17 \( 1 - 102.T + 4.91e3T^{2} \)
19 \( 1 - 46.2T + 6.85e3T^{2} \)
23 \( 1 + 99.1T + 1.21e4T^{2} \)
29 \( 1 + 119.T + 2.43e4T^{2} \)
31 \( 1 - 20.2T + 2.97e4T^{2} \)
37 \( 1 - 117.T + 5.06e4T^{2} \)
41 \( 1 + 278.T + 6.89e4T^{2} \)
43 \( 1 + 484.T + 7.95e4T^{2} \)
47 \( 1 + 347.T + 1.03e5T^{2} \)
53 \( 1 - 161.T + 1.48e5T^{2} \)
61 \( 1 + 845.T + 2.26e5T^{2} \)
67 \( 1 + 740.T + 3.00e5T^{2} \)
71 \( 1 + 738.T + 3.57e5T^{2} \)
73 \( 1 - 539.T + 3.89e5T^{2} \)
79 \( 1 + 412.T + 4.93e5T^{2} \)
83 \( 1 + 75.5T + 5.71e5T^{2} \)
89 \( 1 - 163.T + 7.04e5T^{2} \)
97 \( 1 - 857.T + 9.12e5T^{2} \)
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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

−11.64981815872381445976106930478, −10.04693657925719263437152223942, −9.904059443552255887707084344978, −9.069492869203065510654300770125, −7.61357975836101122598688851698, −6.59631982722821330135637840485, −5.93170950683759556616019645673, −3.36500429690926176582058097010, −1.50395635050927574200838444523, 0, 1.50395635050927574200838444523, 3.36500429690926176582058097010, 5.93170950683759556616019645673, 6.59631982722821330135637840485, 7.61357975836101122598688851698, 9.069492869203065510654300770125, 9.904059443552255887707084344978, 10.04693657925719263437152223942, 11.64981815872381445976106930478

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