Properties

Label 2-177-1.1-c5-0-15
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.29·2-s − 9·3-s + 7.57·4-s + 82.0·5-s + 56.6·6-s + 51.5·7-s + 153.·8-s + 81·9-s − 516.·10-s + 677.·11-s − 68.1·12-s + 290.·13-s − 324.·14-s − 738.·15-s − 1.20e3·16-s + 1.79e3·17-s − 509.·18-s − 766.·19-s + 621.·20-s − 463.·21-s − 4.26e3·22-s − 3.88e3·23-s − 1.38e3·24-s + 3.61e3·25-s − 1.82e3·26-s − 729·27-s + 390.·28-s + ⋯
L(s)  = 1  − 1.11·2-s − 0.577·3-s + 0.236·4-s + 1.46·5-s + 0.642·6-s + 0.397·7-s + 0.848·8-s + 0.333·9-s − 1.63·10-s + 1.68·11-s − 0.136·12-s + 0.477·13-s − 0.442·14-s − 0.847·15-s − 1.18·16-s + 1.50·17-s − 0.370·18-s − 0.487·19-s + 0.347·20-s − 0.229·21-s − 1.87·22-s − 1.52·23-s − 0.490·24-s + 1.15·25-s − 0.530·26-s − 0.192·27-s + 0.0941·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.384053701\)
\(L(\frac12)\) \(\approx\) \(1.384053701\)
\(L(\frac{7}{2})\) 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 \( 1 + 6.29T + 32T^{2} \)
5 \( 1 - 82.0T + 3.12e3T^{2} \)
7 \( 1 - 51.5T + 1.68e4T^{2} \)
11 \( 1 - 677.T + 1.61e5T^{2} \)
13 \( 1 - 290.T + 3.71e5T^{2} \)
17 \( 1 - 1.79e3T + 1.41e6T^{2} \)
19 \( 1 + 766.T + 2.47e6T^{2} \)
23 \( 1 + 3.88e3T + 6.43e6T^{2} \)
29 \( 1 + 4.72e3T + 2.05e7T^{2} \)
31 \( 1 - 9.41e3T + 2.86e7T^{2} \)
37 \( 1 + 1.02e4T + 6.93e7T^{2} \)
41 \( 1 - 1.30e4T + 1.15e8T^{2} \)
43 \( 1 - 2.17e4T + 1.47e8T^{2} \)
47 \( 1 - 3.79e3T + 2.29e8T^{2} \)
53 \( 1 + 5.74e3T + 4.18e8T^{2} \)
61 \( 1 + 3.45e4T + 8.44e8T^{2} \)
67 \( 1 - 4.86e4T + 1.35e9T^{2} \)
71 \( 1 - 4.01e4T + 1.80e9T^{2} \)
73 \( 1 + 2.28e4T + 2.07e9T^{2} \)
79 \( 1 + 2.30e4T + 3.07e9T^{2} \)
83 \( 1 + 6.67e4T + 3.93e9T^{2} \)
89 \( 1 + 4.03e4T + 5.58e9T^{2} \)
97 \( 1 - 7.50e4T + 8.58e9T^{2} \)
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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

−11.55800941665129242693644783631, −10.44895498239637623489998963161, −9.748385965279275943176905707441, −9.049170209256535329814329083858, −7.86845626215502830910492757417, −6.48944091338860879891763753306, −5.67235855611221371363276848053, −4.16494482603061478615931052021, −1.80591707110677906970293135019, −1.01810232019338471843676685262, 1.01810232019338471843676685262, 1.80591707110677906970293135019, 4.16494482603061478615931052021, 5.67235855611221371363276848053, 6.48944091338860879891763753306, 7.86845626215502830910492757417, 9.049170209256535329814329083858, 9.748385965279275943176905707441, 10.44895498239637623489998963161, 11.55800941665129242693644783631

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