Properties

Label 2-177-1.1-c3-0-26
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3.39·2-s − 3·3-s + 3.52·4-s − 6.09·5-s − 10.1·6-s + 1.40·7-s − 15.1·8-s + 9·9-s − 20.6·10-s + 1.38·11-s − 10.5·12-s − 63.1·13-s + 4.76·14-s + 18.2·15-s − 79.7·16-s − 121.·17-s + 30.5·18-s − 44.5·19-s − 21.4·20-s − 4.20·21-s + 4.69·22-s + 138.·23-s + 45.5·24-s − 87.8·25-s − 214.·26-s − 27·27-s + 4.94·28-s + ⋯
L(s)  = 1  + 1.20·2-s − 0.577·3-s + 0.440·4-s − 0.545·5-s − 0.692·6-s + 0.0757·7-s − 0.671·8-s + 0.333·9-s − 0.654·10-s + 0.0378·11-s − 0.254·12-s − 1.34·13-s + 0.0909·14-s + 0.314·15-s − 1.24·16-s − 1.72·17-s + 0.400·18-s − 0.537·19-s − 0.240·20-s − 0.0437·21-s + 0.0454·22-s + 1.25·23-s + 0.387·24-s − 0.702·25-s − 1.61·26-s − 0.192·27-s + 0.0333·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
59 \( 1 - 59T \)
good2 \( 1 - 3.39T + 8T^{2} \)
5 \( 1 + 6.09T + 125T^{2} \)
7 \( 1 - 1.40T + 343T^{2} \)
11 \( 1 - 1.38T + 1.33e3T^{2} \)
13 \( 1 + 63.1T + 2.19e3T^{2} \)
17 \( 1 + 121.T + 4.91e3T^{2} \)
19 \( 1 + 44.5T + 6.85e3T^{2} \)
23 \( 1 - 138.T + 1.21e4T^{2} \)
29 \( 1 - 6.31T + 2.43e4T^{2} \)
31 \( 1 - 246.T + 2.97e4T^{2} \)
37 \( 1 - 300.T + 5.06e4T^{2} \)
41 \( 1 - 110.T + 6.89e4T^{2} \)
43 \( 1 + 156.T + 7.95e4T^{2} \)
47 \( 1 + 338.T + 1.03e5T^{2} \)
53 \( 1 - 484.T + 1.48e5T^{2} \)
61 \( 1 - 606.T + 2.26e5T^{2} \)
67 \( 1 + 724.T + 3.00e5T^{2} \)
71 \( 1 + 345.T + 3.57e5T^{2} \)
73 \( 1 + 1.15e3T + 3.89e5T^{2} \)
79 \( 1 + 145.T + 4.93e5T^{2} \)
83 \( 1 + 858.T + 5.71e5T^{2} \)
89 \( 1 - 6.88T + 7.04e5T^{2} \)
97 \( 1 + 540.T + 9.12e5T^{2} \)
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

−11.79714379223801896226705076279, −11.28139985205193439746675202113, −9.879210385815398916744400220468, −8.644803837501374254809220514966, −7.16773300611028429954840923658, −6.21312739133989569841835040733, −4.86399705481368748169217193795, −4.29183320713496553631172998655, −2.64189293885785430780595029057, 0, 2.64189293885785430780595029057, 4.29183320713496553631172998655, 4.86399705481368748169217193795, 6.21312739133989569841835040733, 7.16773300611028429954840923658, 8.644803837501374254809220514966, 9.879210385815398916744400220468, 11.28139985205193439746675202113, 11.79714379223801896226705076279

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