Properties

Label 2-177-1.1-c13-0-37
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $189.798$
Root an. cond. $13.7767$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 152.·2-s + 729·3-s + 1.49e4·4-s − 4.25e4·5-s − 1.10e5·6-s − 5.71e5·7-s − 1.03e6·8-s + 5.31e5·9-s + 6.48e6·10-s − 6.58e6·11-s + 1.09e7·12-s + 4.28e6·13-s + 8.69e7·14-s − 3.10e7·15-s + 3.44e7·16-s − 1.87e8·17-s − 8.08e7·18-s − 2.31e8·19-s − 6.37e8·20-s − 4.16e8·21-s + 1.00e9·22-s + 1.28e9·23-s − 7.52e8·24-s + 5.92e8·25-s − 6.52e8·26-s + 3.87e8·27-s − 8.55e9·28-s + ⋯
L(s)  = 1  − 1.68·2-s + 0.577·3-s + 1.82·4-s − 1.21·5-s − 0.970·6-s − 1.83·7-s − 1.39·8-s + 0.333·9-s + 2.04·10-s − 1.12·11-s + 1.05·12-s + 0.246·13-s + 3.08·14-s − 0.703·15-s + 0.513·16-s − 1.88·17-s − 0.560·18-s − 1.13·19-s − 2.22·20-s − 1.06·21-s + 1.88·22-s + 1.81·23-s − 0.804·24-s + 0.485·25-s − 0.413·26-s + 0.192·27-s − 3.35·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(189.798\)
Root analytic conductor: \(13.7767\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 729T \)
59 \( 1 - 4.21e10T \)
good2 \( 1 + 152.T + 8.19e3T^{2} \)
5 \( 1 + 4.25e4T + 1.22e9T^{2} \)
7 \( 1 + 5.71e5T + 9.68e10T^{2} \)
11 \( 1 + 6.58e6T + 3.45e13T^{2} \)
13 \( 1 - 4.28e6T + 3.02e14T^{2} \)
17 \( 1 + 1.87e8T + 9.90e15T^{2} \)
19 \( 1 + 2.31e8T + 4.20e16T^{2} \)
23 \( 1 - 1.28e9T + 5.04e17T^{2} \)
29 \( 1 - 2.90e9T + 1.02e19T^{2} \)
31 \( 1 + 1.60e9T + 2.44e19T^{2} \)
37 \( 1 - 2.91e10T + 2.43e20T^{2} \)
41 \( 1 + 3.68e10T + 9.25e20T^{2} \)
43 \( 1 - 2.91e10T + 1.71e21T^{2} \)
47 \( 1 + 2.29e10T + 5.46e21T^{2} \)
53 \( 1 + 1.99e11T + 2.60e22T^{2} \)
61 \( 1 - 4.98e11T + 1.61e23T^{2} \)
67 \( 1 - 3.43e11T + 5.48e23T^{2} \)
71 \( 1 - 1.13e12T + 1.16e24T^{2} \)
73 \( 1 + 1.48e12T + 1.67e24T^{2} \)
79 \( 1 - 2.64e12T + 4.66e24T^{2} \)
83 \( 1 + 3.22e12T + 8.87e24T^{2} \)
89 \( 1 + 6.12e12T + 2.19e25T^{2} \)
97 \( 1 - 7.26e11T + 6.73e25T^{2} \)
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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

−9.614063287223889507770844659732, −8.809946159372137245978627345453, −8.145765358454272451894373754155, −7.05976233800986606701784440658, −6.52007972042882381908814240557, −4.36121222594923517553323856013, −3.07647151600623654293445040679, −2.36996870818998476553097702891, −0.64447348473997649911969013082, 0, 0.64447348473997649911969013082, 2.36996870818998476553097702891, 3.07647151600623654293445040679, 4.36121222594923517553323856013, 6.52007972042882381908814240557, 7.05976233800986606701784440658, 8.145765358454272451894373754155, 8.809946159372137245978627345453, 9.614063287223889507770844659732

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