Properties

Label 2-177-1.1-c5-0-20
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  + 0.290·2-s + 9·3-s − 31.9·4-s + 87.1·5-s + 2.61·6-s + 167.·7-s − 18.5·8-s + 81·9-s + 25.2·10-s + 252.·11-s − 287.·12-s − 289.·13-s + 48.6·14-s + 783.·15-s + 1.01e3·16-s − 227.·17-s + 23.4·18-s − 1.16e3·19-s − 2.78e3·20-s + 1.50e3·21-s + 73.3·22-s − 1.23e3·23-s − 166.·24-s + 4.46e3·25-s − 83.9·26-s + 729·27-s − 5.34e3·28-s + ⋯
L(s)  = 1  + 0.0512·2-s + 0.577·3-s − 0.997·4-s + 1.55·5-s + 0.0296·6-s + 1.29·7-s − 0.102·8-s + 0.333·9-s + 0.0799·10-s + 0.629·11-s − 0.575·12-s − 0.474·13-s + 0.0662·14-s + 0.899·15-s + 0.992·16-s − 0.190·17-s + 0.0170·18-s − 0.738·19-s − 1.55·20-s + 0.746·21-s + 0.0323·22-s − 0.485·23-s − 0.0591·24-s + 1.42·25-s − 0.0243·26-s + 0.192·27-s − 1.28·28-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

\(L(3)\) \(\approx\) \(3.112503158\)
\(L(\frac12)\) \(\approx\) \(3.112503158\)
\(L(\frac{7}{2})\) 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 \( 1 - 0.290T + 32T^{2} \)
5 \( 1 - 87.1T + 3.12e3T^{2} \)
7 \( 1 - 167.T + 1.68e4T^{2} \)
11 \( 1 - 252.T + 1.61e5T^{2} \)
13 \( 1 + 289.T + 3.71e5T^{2} \)
17 \( 1 + 227.T + 1.41e6T^{2} \)
19 \( 1 + 1.16e3T + 2.47e6T^{2} \)
23 \( 1 + 1.23e3T + 6.43e6T^{2} \)
29 \( 1 - 1.97e3T + 2.05e7T^{2} \)
31 \( 1 - 7.79e3T + 2.86e7T^{2} \)
37 \( 1 - 7.04e3T + 6.93e7T^{2} \)
41 \( 1 + 1.12e4T + 1.15e8T^{2} \)
43 \( 1 - 1.85e4T + 1.47e8T^{2} \)
47 \( 1 + 2.01e4T + 2.29e8T^{2} \)
53 \( 1 - 3.73e4T + 4.18e8T^{2} \)
61 \( 1 + 998.T + 8.44e8T^{2} \)
67 \( 1 + 1.96e4T + 1.35e9T^{2} \)
71 \( 1 - 2.16e4T + 1.80e9T^{2} \)
73 \( 1 - 4.62e4T + 2.07e9T^{2} \)
79 \( 1 + 8.50e4T + 3.07e9T^{2} \)
83 \( 1 - 7.54e4T + 3.93e9T^{2} \)
89 \( 1 - 1.05e5T + 5.58e9T^{2} \)
97 \( 1 + 1.09e5T + 8.58e9T^{2} \)
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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

−11.94957383148589325517568654644, −10.47050920996282805680145543981, −9.681465861749473721689752122374, −8.831332089985402727810564841659, −8.010804132379282906951652328551, −6.38407365787007087625432191740, −5.17028757549476092791269257381, −4.26189064364207279541971671456, −2.38463552422994502808857675954, −1.23781975604350924910722443042, 1.23781975604350924910722443042, 2.38463552422994502808857675954, 4.26189064364207279541971671456, 5.17028757549476092791269257381, 6.38407365787007087625432191740, 8.010804132379282906951652328551, 8.831332089985402727810564841659, 9.681465861749473721689752122374, 10.47050920996282805680145543981, 11.94957383148589325517568654644

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