Properties

Label 2-177-1.1-c5-0-14
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  − 8.80·2-s − 9·3-s + 45.5·4-s − 88.4·5-s + 79.2·6-s − 134.·7-s − 119.·8-s + 81·9-s + 779.·10-s + 423.·11-s − 410.·12-s − 713.·13-s + 1.18e3·14-s + 796.·15-s − 405.·16-s + 924.·17-s − 713.·18-s − 64.3·19-s − 4.03e3·20-s + 1.20e3·21-s − 3.72e3·22-s + 3.70e3·23-s + 1.07e3·24-s + 4.70e3·25-s + 6.28e3·26-s − 729·27-s − 6.11e3·28-s + ⋯
L(s)  = 1  − 1.55·2-s − 0.577·3-s + 1.42·4-s − 1.58·5-s + 0.898·6-s − 1.03·7-s − 0.660·8-s + 0.333·9-s + 2.46·10-s + 1.05·11-s − 0.822·12-s − 1.17·13-s + 1.61·14-s + 0.913·15-s − 0.396·16-s + 0.775·17-s − 0.518·18-s − 0.0408·19-s − 2.25·20-s + 0.598·21-s − 1.64·22-s + 1.45·23-s + 0.381·24-s + 1.50·25-s + 1.82·26-s − 0.192·27-s − 1.47·28-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 \( 1 + 8.80T + 32T^{2} \)
5 \( 1 + 88.4T + 3.12e3T^{2} \)
7 \( 1 + 134.T + 1.68e4T^{2} \)
11 \( 1 - 423.T + 1.61e5T^{2} \)
13 \( 1 + 713.T + 3.71e5T^{2} \)
17 \( 1 - 924.T + 1.41e6T^{2} \)
19 \( 1 + 64.3T + 2.47e6T^{2} \)
23 \( 1 - 3.70e3T + 6.43e6T^{2} \)
29 \( 1 - 4.99e3T + 2.05e7T^{2} \)
31 \( 1 - 3.09e3T + 2.86e7T^{2} \)
37 \( 1 + 4.97e3T + 6.93e7T^{2} \)
41 \( 1 - 996.T + 1.15e8T^{2} \)
43 \( 1 - 7.71e3T + 1.47e8T^{2} \)
47 \( 1 + 1.72e4T + 2.29e8T^{2} \)
53 \( 1 + 3.47e4T + 4.18e8T^{2} \)
61 \( 1 - 3.33e4T + 8.44e8T^{2} \)
67 \( 1 - 5.04e4T + 1.35e9T^{2} \)
71 \( 1 + 1.69e4T + 1.80e9T^{2} \)
73 \( 1 + 8.45e4T + 2.07e9T^{2} \)
79 \( 1 - 9.98e3T + 3.07e9T^{2} \)
83 \( 1 - 1.18e4T + 3.93e9T^{2} \)
89 \( 1 + 8.14e3T + 5.58e9T^{2} \)
97 \( 1 + 1.37e5T + 8.58e9T^{2} \)
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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

−11.22746421713079910423052623075, −10.12761257652347143571527646715, −9.343553493711148424609837371489, −8.255802722054470122605156239958, −7.23750302387349980487019811860, −6.63544486289913354710916066222, −4.62468693294212444771803548278, −3.16413302579209302962500184764, −0.962566036084114351544376178999, 0, 0.962566036084114351544376178999, 3.16413302579209302962500184764, 4.62468693294212444771803548278, 6.63544486289913354710916066222, 7.23750302387349980487019811860, 8.255802722054470122605156239958, 9.343553493711148424609837371489, 10.12761257652347143571527646715, 11.22746421713079910423052623075

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