Properties

Label 2-177-1.1-c5-0-40
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  + 3.20·2-s + 9·3-s − 21.7·4-s + 26.7·5-s + 28.8·6-s + 39.0·7-s − 172.·8-s + 81·9-s + 85.6·10-s − 606.·11-s − 195.·12-s − 161.·13-s + 125.·14-s + 240.·15-s + 142.·16-s − 1.65e3·17-s + 259.·18-s + 882.·19-s − 580.·20-s + 351.·21-s − 1.94e3·22-s + 2.92e3·23-s − 1.55e3·24-s − 2.41e3·25-s − 518.·26-s + 729·27-s − 847.·28-s + ⋯
L(s)  = 1  + 0.566·2-s + 0.577·3-s − 0.678·4-s + 0.478·5-s + 0.327·6-s + 0.301·7-s − 0.951·8-s + 0.333·9-s + 0.270·10-s − 1.51·11-s − 0.391·12-s − 0.265·13-s + 0.170·14-s + 0.276·15-s + 0.139·16-s − 1.38·17-s + 0.188·18-s + 0.560·19-s − 0.324·20-s + 0.173·21-s − 0.856·22-s + 1.15·23-s − 0.549·24-s − 0.771·25-s − 0.150·26-s + 0.192·27-s − 0.204·28-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 \( 1 - 3.20T + 32T^{2} \)
5 \( 1 - 26.7T + 3.12e3T^{2} \)
7 \( 1 - 39.0T + 1.68e4T^{2} \)
11 \( 1 + 606.T + 1.61e5T^{2} \)
13 \( 1 + 161.T + 3.71e5T^{2} \)
17 \( 1 + 1.65e3T + 1.41e6T^{2} \)
19 \( 1 - 882.T + 2.47e6T^{2} \)
23 \( 1 - 2.92e3T + 6.43e6T^{2} \)
29 \( 1 + 7.06e3T + 2.05e7T^{2} \)
31 \( 1 + 2.25e3T + 2.86e7T^{2} \)
37 \( 1 + 7.56e3T + 6.93e7T^{2} \)
41 \( 1 + 1.67e4T + 1.15e8T^{2} \)
43 \( 1 + 4.50e3T + 1.47e8T^{2} \)
47 \( 1 - 8.40e3T + 2.29e8T^{2} \)
53 \( 1 - 1.10e4T + 4.18e8T^{2} \)
61 \( 1 - 3.47e4T + 8.44e8T^{2} \)
67 \( 1 + 5.72e4T + 1.35e9T^{2} \)
71 \( 1 - 26.9T + 1.80e9T^{2} \)
73 \( 1 - 5.44e4T + 2.07e9T^{2} \)
79 \( 1 - 9.31e4T + 3.07e9T^{2} \)
83 \( 1 + 8.96e4T + 3.93e9T^{2} \)
89 \( 1 + 1.28e5T + 5.58e9T^{2} \)
97 \( 1 + 2.66e4T + 8.58e9T^{2} \)
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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

−11.35430046687357032202364537781, −10.17457709440697623104184896636, −9.221183020232604127936133190042, −8.348289444582176558661839750272, −7.15958189318662469081587646001, −5.55433893502766979357090847529, −4.79641795752196435130653842974, −3.39295175934334592423867425948, −2.11915960750502431681863590638, 0, 2.11915960750502431681863590638, 3.39295175934334592423867425948, 4.79641795752196435130653842974, 5.55433893502766979357090847529, 7.15958189318662469081587646001, 8.348289444582176558661839750272, 9.221183020232604127936133190042, 10.17457709440697623104184896636, 11.35430046687357032202364537781

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