Properties

Label 2-177-1.1-c13-0-52
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  − 73.1·2-s − 729·3-s − 2.84e3·4-s − 3.78e4·5-s + 5.33e4·6-s + 3.70e5·7-s + 8.07e5·8-s + 5.31e5·9-s + 2.77e6·10-s − 7.88e6·11-s + 2.07e6·12-s − 2.23e6·13-s − 2.70e7·14-s + 2.76e7·15-s − 3.57e7·16-s + 6.39e7·17-s − 3.88e7·18-s + 1.32e8·19-s + 1.07e8·20-s − 2.69e8·21-s + 5.76e8·22-s − 1.21e9·23-s − 5.88e8·24-s + 2.15e8·25-s + 1.63e8·26-s − 3.87e8·27-s − 1.05e9·28-s + ⋯
L(s)  = 1  − 0.807·2-s − 0.577·3-s − 0.347·4-s − 1.08·5-s + 0.466·6-s + 1.18·7-s + 1.08·8-s + 0.333·9-s + 0.876·10-s − 1.34·11-s + 0.200·12-s − 0.128·13-s − 0.960·14-s + 0.626·15-s − 0.532·16-s + 0.642·17-s − 0.269·18-s + 0.645·19-s + 0.376·20-s − 0.686·21-s + 1.08·22-s − 1.71·23-s − 0.628·24-s + 0.176·25-s + 0.103·26-s − 0.192·27-s − 0.412·28-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 \( 1 + 73.1T + 8.19e3T^{2} \)
5 \( 1 + 3.78e4T + 1.22e9T^{2} \)
7 \( 1 - 3.70e5T + 9.68e10T^{2} \)
11 \( 1 + 7.88e6T + 3.45e13T^{2} \)
13 \( 1 + 2.23e6T + 3.02e14T^{2} \)
17 \( 1 - 6.39e7T + 9.90e15T^{2} \)
19 \( 1 - 1.32e8T + 4.20e16T^{2} \)
23 \( 1 + 1.21e9T + 5.04e17T^{2} \)
29 \( 1 - 4.50e9T + 1.02e19T^{2} \)
31 \( 1 + 6.42e9T + 2.44e19T^{2} \)
37 \( 1 + 2.37e10T + 2.43e20T^{2} \)
41 \( 1 - 2.80e10T + 9.25e20T^{2} \)
43 \( 1 + 2.26e10T + 1.71e21T^{2} \)
47 \( 1 - 6.72e10T + 5.46e21T^{2} \)
53 \( 1 - 1.27e11T + 2.60e22T^{2} \)
61 \( 1 - 7.06e11T + 1.61e23T^{2} \)
67 \( 1 + 9.15e11T + 5.48e23T^{2} \)
71 \( 1 - 1.48e12T + 1.16e24T^{2} \)
73 \( 1 + 1.43e12T + 1.67e24T^{2} \)
79 \( 1 - 9.88e11T + 4.66e24T^{2} \)
83 \( 1 - 7.61e11T + 8.87e24T^{2} \)
89 \( 1 - 4.89e12T + 2.19e25T^{2} \)
97 \( 1 - 1.14e13T + 6.73e25T^{2} \)
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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

−10.05180480598474583184045336530, −8.593699192895657041798103513815, −7.85201972044413462329968888565, −7.40985716969272986705088200977, −5.49647105741499557610868260930, −4.75574681851758811839102802574, −3.74955034912575380136404449462, −1.99669508631415985673779530942, −0.813442862828158527708203291566, 0, 0.813442862828158527708203291566, 1.99669508631415985673779530942, 3.74955034912575380136404449462, 4.75574681851758811839102802574, 5.49647105741499557610868260930, 7.40985716969272986705088200977, 7.85201972044413462329968888565, 8.593699192895657041798103513815, 10.05180480598474583184045336530

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