Properties

Label 2-177-1.1-c5-0-39
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  + 10.4·2-s + 9·3-s + 78.0·4-s + 46.0·5-s + 94.4·6-s + 183.·7-s + 483.·8-s + 81·9-s + 483.·10-s − 536.·11-s + 702.·12-s − 601.·13-s + 1.92e3·14-s + 414.·15-s + 2.57e3·16-s − 1.94e3·17-s + 849.·18-s − 1.82e3·19-s + 3.59e3·20-s + 1.64e3·21-s − 5.62e3·22-s − 143.·23-s + 4.35e3·24-s − 1.00e3·25-s − 6.31e3·26-s + 729·27-s + 1.43e4·28-s + ⋯
L(s)  = 1  + 1.85·2-s + 0.577·3-s + 2.44·4-s + 0.824·5-s + 1.07·6-s + 1.41·7-s + 2.67·8-s + 0.333·9-s + 1.52·10-s − 1.33·11-s + 1.40·12-s − 0.987·13-s + 2.62·14-s + 0.475·15-s + 2.51·16-s − 1.63·17-s + 0.618·18-s − 1.16·19-s + 2.01·20-s + 0.815·21-s − 2.47·22-s − 0.0566·23-s + 1.54·24-s − 0.320·25-s − 1.83·26-s + 0.192·27-s + 3.44·28-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

\(L(3)\) \(\approx\) \(8.633002728\)
\(L(\frac12)\) \(\approx\) \(8.633002728\)
\(L(\frac{7}{2})\) 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 \( 1 - 10.4T + 32T^{2} \)
5 \( 1 - 46.0T + 3.12e3T^{2} \)
7 \( 1 - 183.T + 1.68e4T^{2} \)
11 \( 1 + 536.T + 1.61e5T^{2} \)
13 \( 1 + 601.T + 3.71e5T^{2} \)
17 \( 1 + 1.94e3T + 1.41e6T^{2} \)
19 \( 1 + 1.82e3T + 2.47e6T^{2} \)
23 \( 1 + 143.T + 6.43e6T^{2} \)
29 \( 1 - 4.18e3T + 2.05e7T^{2} \)
31 \( 1 - 6.33e3T + 2.86e7T^{2} \)
37 \( 1 + 1.98e3T + 6.93e7T^{2} \)
41 \( 1 - 1.80e4T + 1.15e8T^{2} \)
43 \( 1 + 1.62e4T + 1.47e8T^{2} \)
47 \( 1 - 1.00e4T + 2.29e8T^{2} \)
53 \( 1 - 1.70e4T + 4.18e8T^{2} \)
61 \( 1 + 1.08e4T + 8.44e8T^{2} \)
67 \( 1 - 5.32e4T + 1.35e9T^{2} \)
71 \( 1 + 8.24e4T + 1.80e9T^{2} \)
73 \( 1 + 2.59e4T + 2.07e9T^{2} \)
79 \( 1 - 1.86e4T + 3.07e9T^{2} \)
83 \( 1 - 1.05e5T + 3.93e9T^{2} \)
89 \( 1 + 6.14e4T + 5.58e9T^{2} \)
97 \( 1 - 1.70e5T + 8.58e9T^{2} \)
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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

−12.11777636748009842954853362535, −11.02737081007042308086036026653, −10.24328135833529243344313787390, −8.461776123085412903249320910277, −7.38683189863956555657663459835, −6.17694478639302551874156748635, −4.95315966675852746144470160854, −4.44173186132094496042242604062, −2.51246152752769520170686628833, −2.09147037149266073019168229759, 2.09147037149266073019168229759, 2.51246152752769520170686628833, 4.44173186132094496042242604062, 4.95315966675852746144470160854, 6.17694478639302551874156748635, 7.38683189863956555657663459835, 8.461776123085412903249320910277, 10.24328135833529243344313787390, 11.02737081007042308086036026653, 12.11777636748009842954853362535

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