Properties

Label 2-177-1.1-c7-0-39
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  − 19.0·2-s − 27·3-s + 234.·4-s + 37.6·5-s + 513.·6-s + 1.15e3·7-s − 2.02e3·8-s + 729·9-s − 715.·10-s + 1.61e3·11-s − 6.32e3·12-s − 7.20e3·13-s − 2.20e4·14-s − 1.01e3·15-s + 8.48e3·16-s + 2.58e4·17-s − 1.38e4·18-s − 4.59e4·19-s + 8.80e3·20-s − 3.12e4·21-s − 3.07e4·22-s − 5.30e4·23-s + 5.45e4·24-s − 7.67e4·25-s + 1.37e5·26-s − 1.96e4·27-s + 2.71e5·28-s + ⋯
L(s)  = 1  − 1.68·2-s − 0.577·3-s + 1.82·4-s + 0.134·5-s + 0.971·6-s + 1.27·7-s − 1.39·8-s + 0.333·9-s − 0.226·10-s + 0.366·11-s − 1.05·12-s − 0.909·13-s − 2.14·14-s − 0.0776·15-s + 0.517·16-s + 1.27·17-s − 0.560·18-s − 1.53·19-s + 0.246·20-s − 0.737·21-s − 0.616·22-s − 0.909·23-s + 0.805·24-s − 0.981·25-s + 1.52·26-s − 0.192·27-s + 2.33·28-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 \( 1 + 19.0T + 128T^{2} \)
5 \( 1 - 37.6T + 7.81e4T^{2} \)
7 \( 1 - 1.15e3T + 8.23e5T^{2} \)
11 \( 1 - 1.61e3T + 1.94e7T^{2} \)
13 \( 1 + 7.20e3T + 6.27e7T^{2} \)
17 \( 1 - 2.58e4T + 4.10e8T^{2} \)
19 \( 1 + 4.59e4T + 8.93e8T^{2} \)
23 \( 1 + 5.30e4T + 3.40e9T^{2} \)
29 \( 1 - 1.64e5T + 1.72e10T^{2} \)
31 \( 1 - 6.92e4T + 2.75e10T^{2} \)
37 \( 1 + 3.82e5T + 9.49e10T^{2} \)
41 \( 1 + 5.31e5T + 1.94e11T^{2} \)
43 \( 1 - 5.62e5T + 2.71e11T^{2} \)
47 \( 1 - 1.23e6T + 5.06e11T^{2} \)
53 \( 1 - 2.73e5T + 1.17e12T^{2} \)
61 \( 1 - 2.63e6T + 3.14e12T^{2} \)
67 \( 1 + 4.66e6T + 6.06e12T^{2} \)
71 \( 1 + 5.88e6T + 9.09e12T^{2} \)
73 \( 1 + 3.00e6T + 1.10e13T^{2} \)
79 \( 1 - 5.62e6T + 1.92e13T^{2} \)
83 \( 1 + 5.80e6T + 2.71e13T^{2} \)
89 \( 1 + 5.58e6T + 4.42e13T^{2} \)
97 \( 1 - 3.69e6T + 8.07e13T^{2} \)
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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

−10.48005982371747353384531536520, −10.10853388954172458612543742556, −8.777984110478610110036030680619, −7.982069544446889987377687074877, −7.10263364593486627267060474946, −5.83640310317645624415399262905, −4.44417509735790019523278500247, −2.20026742264310832307499913840, −1.25890634735935087849180963810, 0, 1.25890634735935087849180963810, 2.20026742264310832307499913840, 4.44417509735790019523278500247, 5.83640310317645624415399262905, 7.10263364593486627267060474946, 7.982069544446889987377687074877, 8.777984110478610110036030680619, 10.10853388954172458612543742556, 10.48005982371747353384531536520

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