Properties

Label 2-177-1.1-c7-0-58
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  + 14.0·2-s − 27·3-s + 69.6·4-s + 153.·5-s − 379.·6-s − 215.·7-s − 819.·8-s + 729·9-s + 2.15e3·10-s + 7.86e3·11-s − 1.88e3·12-s − 1.42e4·13-s − 3.02e3·14-s − 4.13e3·15-s − 2.04e4·16-s − 1.44e4·17-s + 1.02e4·18-s + 5.22e4·19-s + 1.06e4·20-s + 5.81e3·21-s + 1.10e5·22-s − 2.09e4·23-s + 2.21e4·24-s − 5.46e4·25-s − 2.00e5·26-s − 1.96e4·27-s − 1.49e4·28-s + ⋯
L(s)  = 1  + 1.24·2-s − 0.577·3-s + 0.544·4-s + 0.548·5-s − 0.717·6-s − 0.237·7-s − 0.566·8-s + 0.333·9-s + 0.681·10-s + 1.78·11-s − 0.314·12-s − 1.79·13-s − 0.294·14-s − 0.316·15-s − 1.24·16-s − 0.712·17-s + 0.414·18-s + 1.74·19-s + 0.298·20-s + 0.136·21-s + 2.21·22-s − 0.358·23-s + 0.326·24-s − 0.699·25-s − 2.23·26-s − 0.192·27-s − 0.129·28-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 \( 1 - 14.0T + 128T^{2} \)
5 \( 1 - 153.T + 7.81e4T^{2} \)
7 \( 1 + 215.T + 8.23e5T^{2} \)
11 \( 1 - 7.86e3T + 1.94e7T^{2} \)
13 \( 1 + 1.42e4T + 6.27e7T^{2} \)
17 \( 1 + 1.44e4T + 4.10e8T^{2} \)
19 \( 1 - 5.22e4T + 8.93e8T^{2} \)
23 \( 1 + 2.09e4T + 3.40e9T^{2} \)
29 \( 1 + 1.36e5T + 1.72e10T^{2} \)
31 \( 1 - 9.32e3T + 2.75e10T^{2} \)
37 \( 1 + 5.25e5T + 9.49e10T^{2} \)
41 \( 1 - 1.97e5T + 1.94e11T^{2} \)
43 \( 1 - 4.10e5T + 2.71e11T^{2} \)
47 \( 1 + 9.46e5T + 5.06e11T^{2} \)
53 \( 1 + 1.55e6T + 1.17e12T^{2} \)
61 \( 1 + 6.59e5T + 3.14e12T^{2} \)
67 \( 1 + 2.30e6T + 6.06e12T^{2} \)
71 \( 1 + 1.68e6T + 9.09e12T^{2} \)
73 \( 1 - 2.53e6T + 1.10e13T^{2} \)
79 \( 1 + 1.26e6T + 1.92e13T^{2} \)
83 \( 1 - 2.22e6T + 2.71e13T^{2} \)
89 \( 1 + 1.04e7T + 4.42e13T^{2} \)
97 \( 1 - 5.41e6T + 8.07e13T^{2} \)
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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

−11.43714455181307135785403315325, −9.734028049383961461855538366699, −9.292470125426036328304410072083, −7.26648748625603133986393825718, −6.35949579956293301455957179765, −5.40776231945812800620240659930, −4.48365868999037253929518855936, −3.31760936034376476525492385177, −1.79225551904875205780366645203, 0, 1.79225551904875205780366645203, 3.31760936034376476525492385177, 4.48365868999037253929518855936, 5.40776231945812800620240659930, 6.35949579956293301455957179765, 7.26648748625603133986393825718, 9.292470125426036328304410072083, 9.734028049383961461855538366699, 11.43714455181307135785403315325

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