Properties

Label 2-177-1.1-c3-0-29
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  + 3.44·2-s + 3·3-s + 3.86·4-s − 20.9·5-s + 10.3·6-s − 30.0·7-s − 14.2·8-s + 9·9-s − 72.0·10-s + 51.2·11-s + 11.5·12-s − 22.3·13-s − 103.·14-s − 62.7·15-s − 79.9·16-s − 89.1·17-s + 30.9·18-s + 96.6·19-s − 80.7·20-s − 90.1·21-s + 176.·22-s + 76.1·23-s − 42.7·24-s + 312.·25-s − 76.8·26-s + 27·27-s − 116.·28-s + ⋯
L(s)  = 1  + 1.21·2-s + 0.577·3-s + 0.482·4-s − 1.86·5-s + 0.703·6-s − 1.62·7-s − 0.629·8-s + 0.333·9-s − 2.27·10-s + 1.40·11-s + 0.278·12-s − 0.475·13-s − 1.97·14-s − 1.07·15-s − 1.24·16-s − 1.27·17-s + 0.405·18-s + 1.16·19-s − 0.902·20-s − 0.937·21-s + 1.70·22-s + 0.690·23-s − 0.363·24-s + 2.49·25-s − 0.579·26-s + 0.192·27-s − 0.783·28-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 \( 1 - 3.44T + 8T^{2} \)
5 \( 1 + 20.9T + 125T^{2} \)
7 \( 1 + 30.0T + 343T^{2} \)
11 \( 1 - 51.2T + 1.33e3T^{2} \)
13 \( 1 + 22.3T + 2.19e3T^{2} \)
17 \( 1 + 89.1T + 4.91e3T^{2} \)
19 \( 1 - 96.6T + 6.85e3T^{2} \)
23 \( 1 - 76.1T + 1.21e4T^{2} \)
29 \( 1 + 71.5T + 2.43e4T^{2} \)
31 \( 1 + 129.T + 2.97e4T^{2} \)
37 \( 1 + 108.T + 5.06e4T^{2} \)
41 \( 1 + 357.T + 6.89e4T^{2} \)
43 \( 1 + 237.T + 7.95e4T^{2} \)
47 \( 1 - 97.2T + 1.03e5T^{2} \)
53 \( 1 + 705.T + 1.48e5T^{2} \)
61 \( 1 + 549.T + 2.26e5T^{2} \)
67 \( 1 - 652.T + 3.00e5T^{2} \)
71 \( 1 + 37.3T + 3.57e5T^{2} \)
73 \( 1 + 600.T + 3.89e5T^{2} \)
79 \( 1 - 1.22e3T + 4.93e5T^{2} \)
83 \( 1 + 577.T + 5.71e5T^{2} \)
89 \( 1 + 375.T + 7.04e5T^{2} \)
97 \( 1 - 322.T + 9.12e5T^{2} \)
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

−12.06830654466550296603372280610, −11.25436870884496250578815632714, −9.475517779132627901091902130011, −8.777443380280413713503355129702, −7.20413733869544528667346888973, −6.55840939306120944740231332878, −4.71704285098030707407427347265, −3.64654500629993182497750968523, −3.23020095222386699072236285240, 0, 3.23020095222386699072236285240, 3.64654500629993182497750968523, 4.71704285098030707407427347265, 6.55840939306120944740231332878, 7.20413733869544528667346888973, 8.777443380280413713503355129702, 9.475517779132627901091902130011, 11.25436870884496250578815632714, 12.06830654466550296603372280610

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