Properties

Label 2-177-1.1-c13-0-92
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  − 28.5·2-s + 729·3-s − 7.37e3·4-s − 1.38e4·5-s − 2.07e4·6-s + 3.51e5·7-s + 4.43e5·8-s + 5.31e5·9-s + 3.95e5·10-s + 8.63e6·11-s − 5.37e6·12-s − 2.28e7·13-s − 1.00e7·14-s − 1.01e7·15-s + 4.77e7·16-s − 1.05e6·17-s − 1.51e7·18-s − 3.21e7·19-s + 1.02e8·20-s + 2.56e8·21-s − 2.46e8·22-s − 5.53e8·23-s + 3.23e8·24-s − 1.02e9·25-s + 6.52e8·26-s + 3.87e8·27-s − 2.59e9·28-s + ⋯
L(s)  = 1  − 0.314·2-s + 0.577·3-s − 0.900·4-s − 0.397·5-s − 0.181·6-s + 1.12·7-s + 0.598·8-s + 0.333·9-s + 0.125·10-s + 1.46·11-s − 0.520·12-s − 1.31·13-s − 0.355·14-s − 0.229·15-s + 0.712·16-s − 0.0105·17-s − 0.104·18-s − 0.156·19-s + 0.358·20-s + 0.652·21-s − 0.462·22-s − 0.779·23-s + 0.345·24-s − 0.842·25-s + 0.414·26-s + 0.192·27-s − 1.01·28-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 \( 1 + 28.5T + 8.19e3T^{2} \)
5 \( 1 + 1.38e4T + 1.22e9T^{2} \)
7 \( 1 - 3.51e5T + 9.68e10T^{2} \)
11 \( 1 - 8.63e6T + 3.45e13T^{2} \)
13 \( 1 + 2.28e7T + 3.02e14T^{2} \)
17 \( 1 + 1.05e6T + 9.90e15T^{2} \)
19 \( 1 + 3.21e7T + 4.20e16T^{2} \)
23 \( 1 + 5.53e8T + 5.04e17T^{2} \)
29 \( 1 - 2.88e8T + 1.02e19T^{2} \)
31 \( 1 + 8.59e9T + 2.44e19T^{2} \)
37 \( 1 + 1.31e10T + 2.43e20T^{2} \)
41 \( 1 - 3.93e10T + 9.25e20T^{2} \)
43 \( 1 - 7.30e10T + 1.71e21T^{2} \)
47 \( 1 - 7.35e10T + 5.46e21T^{2} \)
53 \( 1 - 1.27e11T + 2.60e22T^{2} \)
61 \( 1 + 2.72e11T + 1.61e23T^{2} \)
67 \( 1 - 1.83e11T + 5.48e23T^{2} \)
71 \( 1 + 3.93e11T + 1.16e24T^{2} \)
73 \( 1 - 1.61e10T + 1.67e24T^{2} \)
79 \( 1 + 1.21e12T + 4.66e24T^{2} \)
83 \( 1 - 2.98e12T + 8.87e24T^{2} \)
89 \( 1 + 4.57e12T + 2.19e25T^{2} \)
97 \( 1 - 7.94e12T + 6.73e25T^{2} \)
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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

−9.506488669571070519457884637071, −8.949192141651261678678141466653, −7.905227999123865911867737973841, −7.28163909109363095851764693271, −5.55550331634386407861107646825, −4.35823662326415612673633772144, −3.87249759211717344715550965344, −2.16406220079347355969764904033, −1.20862746197247855056610354277, 0, 1.20862746197247855056610354277, 2.16406220079347355969764904033, 3.87249759211717344715550965344, 4.35823662326415612673633772144, 5.55550331634386407861107646825, 7.28163909109363095851764693271, 7.905227999123865911867737973841, 8.949192141651261678678141466653, 9.506488669571070519457884637071

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