Properties

Label 2-177-1.1-c1-0-5
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $1.41335$
Root an. cond. $1.18884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.61·2-s + 3-s + 4.85·4-s − 3·5-s − 2.61·6-s − 2.38·7-s − 7.47·8-s + 9-s + 7.85·10-s + 3.47·11-s + 4.85·12-s − 6.70·13-s + 6.23·14-s − 3·15-s + 9.85·16-s − 2.61·17-s − 2.61·18-s − 5.85·19-s − 14.5·20-s − 2.38·21-s − 9.09·22-s − 1.38·23-s − 7.47·24-s + 4·25-s + 17.5·26-s + 27-s − 11.5·28-s + ⋯
L(s)  = 1  − 1.85·2-s + 0.577·3-s + 2.42·4-s − 1.34·5-s − 1.06·6-s − 0.900·7-s − 2.64·8-s + 0.333·9-s + 2.48·10-s + 1.04·11-s + 1.40·12-s − 1.86·13-s + 1.66·14-s − 0.774·15-s + 2.46·16-s − 0.634·17-s − 0.617·18-s − 1.34·19-s − 3.25·20-s − 0.519·21-s − 1.93·22-s − 0.288·23-s − 1.52·24-s + 0.800·25-s + 3.44·26-s + 0.192·27-s − 2.18·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
59 \( 1 - T \)
good2 \( 1 + 2.61T + 2T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
7 \( 1 + 2.38T + 7T^{2} \)
11 \( 1 - 3.47T + 11T^{2} \)
13 \( 1 + 6.70T + 13T^{2} \)
17 \( 1 + 2.61T + 17T^{2} \)
19 \( 1 + 5.85T + 19T^{2} \)
23 \( 1 + 1.38T + 23T^{2} \)
29 \( 1 + 4.38T + 29T^{2} \)
31 \( 1 + 2.38T + 31T^{2} \)
37 \( 1 - 5.09T + 37T^{2} \)
41 \( 1 + 6.09T + 41T^{2} \)
43 \( 1 - 12.7T + 43T^{2} \)
47 \( 1 + 4.85T + 47T^{2} \)
53 \( 1 - 5.47T + 53T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + 4.23T + 67T^{2} \)
71 \( 1 - 4.23T + 71T^{2} \)
73 \( 1 - 8.09T + 73T^{2} \)
79 \( 1 + 3T + 79T^{2} \)
83 \( 1 + 1.14T + 83T^{2} \)
89 \( 1 - 6.09T + 89T^{2} \)
97 \( 1 + 9.47T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.92696214469704744571880174187, −10.99184770821395818272835611046, −9.827331182086964785223470418324, −9.180492390656979253391309638881, −8.221589439921848690314643683124, −7.31829646127061690907670913077, −6.62774106369167144361747897254, −3.98321240869532954503946024592, −2.41602178571510672764464498026, 0, 2.41602178571510672764464498026, 3.98321240869532954503946024592, 6.62774106369167144361747897254, 7.31829646127061690907670913077, 8.221589439921848690314643683124, 9.180492390656979253391309638881, 9.827331182086964785223470418324, 10.99184770821395818272835611046, 11.92696214469704744571880174187

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