Properties

Label 2-177-1.1-c5-0-34
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.65·2-s − 9·3-s − 29.2·4-s + 59.9·5-s − 14.9·6-s − 87.2·7-s − 101.·8-s + 81·9-s + 99.4·10-s + 655.·11-s + 263.·12-s + 139.·13-s − 144.·14-s − 539.·15-s + 767.·16-s − 1.51e3·17-s + 134.·18-s + 393.·19-s − 1.75e3·20-s + 785.·21-s + 1.08e3·22-s − 4.74e3·23-s + 914.·24-s + 466.·25-s + 232.·26-s − 729·27-s + 2.55e3·28-s + ⋯
L(s)  = 1  + 0.293·2-s − 0.577·3-s − 0.913·4-s + 1.07·5-s − 0.169·6-s − 0.672·7-s − 0.561·8-s + 0.333·9-s + 0.314·10-s + 1.63·11-s + 0.527·12-s + 0.229·13-s − 0.197·14-s − 0.618·15-s + 0.749·16-s − 1.27·17-s + 0.0977·18-s + 0.250·19-s − 0.979·20-s + 0.388·21-s + 0.479·22-s − 1.87·23-s + 0.324·24-s + 0.149·25-s + 0.0673·26-s − 0.192·27-s + 0.615·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9T \)
59 \( 1 + 3.48e3T \)
good2 \( 1 - 1.65T + 32T^{2} \)
5 \( 1 - 59.9T + 3.12e3T^{2} \)
7 \( 1 + 87.2T + 1.68e4T^{2} \)
11 \( 1 - 655.T + 1.61e5T^{2} \)
13 \( 1 - 139.T + 3.71e5T^{2} \)
17 \( 1 + 1.51e3T + 1.41e6T^{2} \)
19 \( 1 - 393.T + 2.47e6T^{2} \)
23 \( 1 + 4.74e3T + 6.43e6T^{2} \)
29 \( 1 - 2.79e3T + 2.05e7T^{2} \)
31 \( 1 + 1.80e3T + 2.86e7T^{2} \)
37 \( 1 - 6.71e3T + 6.93e7T^{2} \)
41 \( 1 + 1.39e4T + 1.15e8T^{2} \)
43 \( 1 + 1.93e4T + 1.47e8T^{2} \)
47 \( 1 + 1.57e4T + 2.29e8T^{2} \)
53 \( 1 + 2.56e4T + 4.18e8T^{2} \)
61 \( 1 - 1.10e4T + 8.44e8T^{2} \)
67 \( 1 + 3.45e4T + 1.35e9T^{2} \)
71 \( 1 - 4.30e4T + 1.80e9T^{2} \)
73 \( 1 + 3.49e4T + 2.07e9T^{2} \)
79 \( 1 - 9.11e4T + 3.07e9T^{2} \)
83 \( 1 + 1.09e5T + 3.93e9T^{2} \)
89 \( 1 - 1.36e5T + 5.58e9T^{2} \)
97 \( 1 + 1.39e5T + 8.58e9T^{2} \)
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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

−11.48029239219723659230717217167, −9.959753471635539700121014989088, −9.549167733143926208900895008114, −8.490023872608393022492700532429, −6.50950284470743050813168071340, −6.09684394869935379223898342811, −4.73681531580150838531449960476, −3.63027831436270921371114883529, −1.64825052274665905681361866588, 0, 1.64825052274665905681361866588, 3.63027831436270921371114883529, 4.73681531580150838531449960476, 6.09684394869935379223898342811, 6.50950284470743050813168071340, 8.490023872608393022492700532429, 9.549167733143926208900895008114, 9.959753471635539700121014989088, 11.48029239219723659230717217167

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