Properties

Label 2-177-1.1-c7-0-42
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  − 17.9·2-s + 27·3-s + 195.·4-s − 98.9·5-s − 485.·6-s + 159.·7-s − 1.21e3·8-s + 729·9-s + 1.77e3·10-s − 4.88e3·11-s + 5.28e3·12-s + 1.16e4·13-s − 2.86e3·14-s − 2.67e3·15-s − 3.15e3·16-s − 1.04e4·17-s − 1.31e4·18-s + 1.49e4·19-s − 1.93e4·20-s + 4.29e3·21-s + 8.78e4·22-s − 88.0·23-s − 3.28e4·24-s − 6.83e4·25-s − 2.09e5·26-s + 1.96e4·27-s + 3.11e4·28-s + ⋯
L(s)  = 1  − 1.59·2-s + 0.577·3-s + 1.52·4-s − 0.353·5-s − 0.918·6-s + 0.175·7-s − 0.839·8-s + 0.333·9-s + 0.562·10-s − 1.10·11-s + 0.882·12-s + 1.47·13-s − 0.278·14-s − 0.204·15-s − 0.192·16-s − 0.516·17-s − 0.530·18-s + 0.500·19-s − 0.540·20-s + 0.101·21-s + 1.75·22-s − 0.00150·23-s − 0.484·24-s − 0.874·25-s − 2.34·26-s + 0.192·27-s + 0.268·28-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 \( 1 + 17.9T + 128T^{2} \)
5 \( 1 + 98.9T + 7.81e4T^{2} \)
7 \( 1 - 159.T + 8.23e5T^{2} \)
11 \( 1 + 4.88e3T + 1.94e7T^{2} \)
13 \( 1 - 1.16e4T + 6.27e7T^{2} \)
17 \( 1 + 1.04e4T + 4.10e8T^{2} \)
19 \( 1 - 1.49e4T + 8.93e8T^{2} \)
23 \( 1 + 88.0T + 3.40e9T^{2} \)
29 \( 1 + 1.02e5T + 1.72e10T^{2} \)
31 \( 1 + 2.87e5T + 2.75e10T^{2} \)
37 \( 1 - 5.66e5T + 9.49e10T^{2} \)
41 \( 1 - 2.96e5T + 1.94e11T^{2} \)
43 \( 1 - 7.40e5T + 2.71e11T^{2} \)
47 \( 1 - 9.58e5T + 5.06e11T^{2} \)
53 \( 1 + 7.32e5T + 1.17e12T^{2} \)
61 \( 1 - 2.28e6T + 3.14e12T^{2} \)
67 \( 1 + 4.29e6T + 6.06e12T^{2} \)
71 \( 1 + 4.99e6T + 9.09e12T^{2} \)
73 \( 1 - 3.66e6T + 1.10e13T^{2} \)
79 \( 1 - 3.18e6T + 1.92e13T^{2} \)
83 \( 1 + 4.43e6T + 2.71e13T^{2} \)
89 \( 1 + 1.07e7T + 4.42e13T^{2} \)
97 \( 1 + 4.05e6T + 8.07e13T^{2} \)
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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

−10.84774396377019065459182349046, −9.617678365131002178285528582597, −8.865514259424435557710908598912, −7.926936903980230984594773082396, −7.39660415192149137076366009713, −5.86670463531824818777491397174, −4.02864230046241893110756464429, −2.52345365575117057242890934295, −1.30871228754938191317522624811, 0, 1.30871228754938191317522624811, 2.52345365575117057242890934295, 4.02864230046241893110756464429, 5.86670463531824818777491397174, 7.39660415192149137076366009713, 7.926936903980230984594773082396, 8.865514259424435557710908598912, 9.617678365131002178285528582597, 10.84774396377019065459182349046

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