Properties

Label 2-177-1.1-c7-0-18
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  − 4.55·2-s − 27·3-s − 107.·4-s − 540.·5-s + 123.·6-s − 1.23e3·7-s + 1.07e3·8-s + 729·9-s + 2.46e3·10-s + 3.52e3·11-s + 2.89e3·12-s − 4.83e3·13-s + 5.64e3·14-s + 1.45e4·15-s + 8.84e3·16-s + 2.24e3·17-s − 3.32e3·18-s + 5.31e4·19-s + 5.79e4·20-s + 3.34e4·21-s − 1.60e4·22-s − 5.63e4·23-s − 2.89e4·24-s + 2.13e5·25-s + 2.20e4·26-s − 1.96e4·27-s + 1.32e5·28-s + ⋯
L(s)  = 1  − 0.402·2-s − 0.577·3-s − 0.837·4-s − 1.93·5-s + 0.232·6-s − 1.36·7-s + 0.740·8-s + 0.333·9-s + 0.778·10-s + 0.797·11-s + 0.483·12-s − 0.610·13-s + 0.549·14-s + 1.11·15-s + 0.539·16-s + 0.110·17-s − 0.134·18-s + 1.77·19-s + 1.61·20-s + 0.787·21-s − 0.321·22-s − 0.966·23-s − 0.427·24-s + 2.73·25-s + 0.245·26-s − 0.192·27-s + 1.14·28-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 \( 1 + 4.55T + 128T^{2} \)
5 \( 1 + 540.T + 7.81e4T^{2} \)
7 \( 1 + 1.23e3T + 8.23e5T^{2} \)
11 \( 1 - 3.52e3T + 1.94e7T^{2} \)
13 \( 1 + 4.83e3T + 6.27e7T^{2} \)
17 \( 1 - 2.24e3T + 4.10e8T^{2} \)
19 \( 1 - 5.31e4T + 8.93e8T^{2} \)
23 \( 1 + 5.63e4T + 3.40e9T^{2} \)
29 \( 1 + 1.37e5T + 1.72e10T^{2} \)
31 \( 1 + 1.21e5T + 2.75e10T^{2} \)
37 \( 1 + 4.76e5T + 9.49e10T^{2} \)
41 \( 1 - 1.40e5T + 1.94e11T^{2} \)
43 \( 1 - 6.96e5T + 2.71e11T^{2} \)
47 \( 1 - 7.83e5T + 5.06e11T^{2} \)
53 \( 1 - 1.05e6T + 1.17e12T^{2} \)
61 \( 1 - 3.78e5T + 3.14e12T^{2} \)
67 \( 1 + 1.12e6T + 6.06e12T^{2} \)
71 \( 1 - 3.82e6T + 9.09e12T^{2} \)
73 \( 1 + 1.68e6T + 1.10e13T^{2} \)
79 \( 1 - 5.56e6T + 1.92e13T^{2} \)
83 \( 1 + 1.80e6T + 2.71e13T^{2} \)
89 \( 1 - 8.26e6T + 4.42e13T^{2} \)
97 \( 1 + 9.36e6T + 8.07e13T^{2} \)
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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

−10.91879785768879829999743299612, −9.755414909826336575978606300606, −8.989440705573759226353044907497, −7.66212736081745067274006759829, −7.06003983732652805071821901495, −5.43208581459069187747024382233, −4.06798025285441737294379372128, −3.48148723352148063994612398685, −0.78777653555696522991821047370, 0, 0.78777653555696522991821047370, 3.48148723352148063994612398685, 4.06798025285441737294379372128, 5.43208581459069187747024382233, 7.06003983732652805071821901495, 7.66212736081745067274006759829, 8.989440705573759226353044907497, 9.755414909826336575978606300606, 10.91879785768879829999743299612

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