Properties

Label 2-177-1.1-c5-0-23
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  − 8.94·2-s − 9·3-s + 47.9·4-s + 48.3·5-s + 80.4·6-s − 32.5·7-s − 142.·8-s + 81·9-s − 432.·10-s − 607.·11-s − 431.·12-s − 114.·13-s + 291.·14-s − 435.·15-s − 257.·16-s + 1.47e3·17-s − 724.·18-s + 1.19e3·19-s + 2.31e3·20-s + 293.·21-s + 5.43e3·22-s − 815.·23-s + 1.28e3·24-s − 788.·25-s + 1.02e3·26-s − 729·27-s − 1.56e3·28-s + ⋯
L(s)  = 1  − 1.58·2-s − 0.577·3-s + 1.49·4-s + 0.864·5-s + 0.912·6-s − 0.251·7-s − 0.789·8-s + 0.333·9-s − 1.36·10-s − 1.51·11-s − 0.865·12-s − 0.187·13-s + 0.397·14-s − 0.499·15-s − 0.251·16-s + 1.24·17-s − 0.526·18-s + 0.761·19-s + 1.29·20-s + 0.145·21-s + 2.39·22-s − 0.321·23-s + 0.455·24-s − 0.252·25-s + 0.296·26-s − 0.192·27-s − 0.376·28-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 \( 1 + 8.94T + 32T^{2} \)
5 \( 1 - 48.3T + 3.12e3T^{2} \)
7 \( 1 + 32.5T + 1.68e4T^{2} \)
11 \( 1 + 607.T + 1.61e5T^{2} \)
13 \( 1 + 114.T + 3.71e5T^{2} \)
17 \( 1 - 1.47e3T + 1.41e6T^{2} \)
19 \( 1 - 1.19e3T + 2.47e6T^{2} \)
23 \( 1 + 815.T + 6.43e6T^{2} \)
29 \( 1 - 7.05e3T + 2.05e7T^{2} \)
31 \( 1 + 2.01e3T + 2.86e7T^{2} \)
37 \( 1 - 2.88e3T + 6.93e7T^{2} \)
41 \( 1 - 7.97e3T + 1.15e8T^{2} \)
43 \( 1 + 1.17e4T + 1.47e8T^{2} \)
47 \( 1 - 2.21e3T + 2.29e8T^{2} \)
53 \( 1 + 1.68e4T + 4.18e8T^{2} \)
61 \( 1 + 1.13e4T + 8.44e8T^{2} \)
67 \( 1 + 2.93e4T + 1.35e9T^{2} \)
71 \( 1 - 6.18e4T + 1.80e9T^{2} \)
73 \( 1 + 6.95e3T + 2.07e9T^{2} \)
79 \( 1 + 7.44e4T + 3.07e9T^{2} \)
83 \( 1 + 6.48e4T + 3.93e9T^{2} \)
89 \( 1 + 6.47e3T + 5.58e9T^{2} \)
97 \( 1 - 3.47e4T + 8.58e9T^{2} \)
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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

−10.84576262061711510794983542130, −10.02317355665300443082534312856, −9.657514779989861200963392679450, −8.217317167484078890031246792006, −7.42929376208051687004919942623, −6.16411414208882051440013683803, −5.11024244142710189943624276096, −2.72526690044383936691500616946, −1.32417028460337714389596590970, 0, 1.32417028460337714389596590970, 2.72526690044383936691500616946, 5.11024244142710189943624276096, 6.16411414208882051440013683803, 7.42929376208051687004919942623, 8.217317167484078890031246792006, 9.657514779989861200963392679450, 10.02317355665300443082534312856, 10.84576262061711510794983542130

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