Properties

Label 2-177-1.1-c7-0-27
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $55.2921$
Root an. cond. $7.43586$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.97·2-s − 27·3-s − 124.·4-s − 339.·5-s + 53.2·6-s + 364.·7-s + 497.·8-s + 729·9-s + 669.·10-s − 3.58e3·11-s + 3.35e3·12-s + 857.·13-s − 718.·14-s + 9.17e3·15-s + 1.49e4·16-s + 3.64e3·17-s − 1.43e3·18-s − 9.65e3·19-s + 4.21e4·20-s − 9.84e3·21-s + 7.07e3·22-s + 1.16e5·23-s − 1.34e4·24-s + 3.73e4·25-s − 1.69e3·26-s − 1.96e4·27-s − 4.52e4·28-s + ⋯
L(s)  = 1  − 0.174·2-s − 0.577·3-s − 0.969·4-s − 1.21·5-s + 0.100·6-s + 0.401·7-s + 0.343·8-s + 0.333·9-s + 0.211·10-s − 0.812·11-s + 0.559·12-s + 0.108·13-s − 0.0700·14-s + 0.701·15-s + 0.909·16-s + 0.180·17-s − 0.0580·18-s − 0.322·19-s + 1.17·20-s − 0.232·21-s + 0.141·22-s + 1.99·23-s − 0.198·24-s + 0.477·25-s − 0.0188·26-s − 0.192·27-s − 0.389·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(55.2921\)
Root analytic conductor: \(7.43586\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 27T \)
59 \( 1 - 2.05e5T \)
good2 \( 1 + 1.97T + 128T^{2} \)
5 \( 1 + 339.T + 7.81e4T^{2} \)
7 \( 1 - 364.T + 8.23e5T^{2} \)
11 \( 1 + 3.58e3T + 1.94e7T^{2} \)
13 \( 1 - 857.T + 6.27e7T^{2} \)
17 \( 1 - 3.64e3T + 4.10e8T^{2} \)
19 \( 1 + 9.65e3T + 8.93e8T^{2} \)
23 \( 1 - 1.16e5T + 3.40e9T^{2} \)
29 \( 1 - 2.47e5T + 1.72e10T^{2} \)
31 \( 1 - 1.14e5T + 2.75e10T^{2} \)
37 \( 1 + 1.94e5T + 9.49e10T^{2} \)
41 \( 1 + 5.94e5T + 1.94e11T^{2} \)
43 \( 1 - 2.47e5T + 2.71e11T^{2} \)
47 \( 1 + 1.30e6T + 5.06e11T^{2} \)
53 \( 1 - 1.20e6T + 1.17e12T^{2} \)
61 \( 1 - 4.28e5T + 3.14e12T^{2} \)
67 \( 1 + 2.01e6T + 6.06e12T^{2} \)
71 \( 1 - 1.87e6T + 9.09e12T^{2} \)
73 \( 1 - 2.35e6T + 1.10e13T^{2} \)
79 \( 1 + 3.74e6T + 1.92e13T^{2} \)
83 \( 1 - 3.21e6T + 2.71e13T^{2} \)
89 \( 1 + 5.44e5T + 4.42e13T^{2} \)
97 \( 1 + 2.91e6T + 8.07e13T^{2} \)
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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

−10.90526341917156237298952497273, −10.03268613288349471998174241960, −8.614067548044543250000034823802, −8.010298518558935259094548783402, −6.84358754885298385479482981122, −5.15430608285939372847791262039, −4.54875424800298307180390953584, −3.23291480656702310762078093981, −1.02251000544528177480807705273, 0, 1.02251000544528177480807705273, 3.23291480656702310762078093981, 4.54875424800298307180390953584, 5.15430608285939372847791262039, 6.84358754885298385479482981122, 8.010298518558935259094548783402, 8.614067548044543250000034823802, 10.03268613288349471998174241960, 10.90526341917156237298952497273

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