Properties

Label 2-177-1.1-c3-0-8
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 $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.26·2-s + 3·3-s − 6.38·4-s + 14.8·5-s − 3.80·6-s + 22.4·7-s + 18.2·8-s + 9·9-s − 18.8·10-s − 70.2·11-s − 19.1·12-s − 0.125·13-s − 28.5·14-s + 44.6·15-s + 27.9·16-s + 57.2·17-s − 11.4·18-s + 40.8·19-s − 95.0·20-s + 67.4·21-s + 89.1·22-s + 190.·23-s + 54.7·24-s + 96.4·25-s + 0.159·26-s + 27·27-s − 143.·28-s + ⋯
L(s)  = 1  − 0.448·2-s + 0.577·3-s − 0.798·4-s + 1.33·5-s − 0.259·6-s + 1.21·7-s + 0.807·8-s + 0.333·9-s − 0.597·10-s − 1.92·11-s − 0.461·12-s − 0.00267·13-s − 0.544·14-s + 0.768·15-s + 0.436·16-s + 0.817·17-s − 0.149·18-s + 0.492·19-s − 1.06·20-s + 0.701·21-s + 0.863·22-s + 1.72·23-s + 0.465·24-s + 0.771·25-s + 0.00119·26-s + 0.192·27-s − 0.969·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.888605372\)
\(L(\frac12)\) \(\approx\) \(1.888605372\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.26T + 8T^{2} \)
5 \( 1 - 14.8T + 125T^{2} \)
7 \( 1 - 22.4T + 343T^{2} \)
11 \( 1 + 70.2T + 1.33e3T^{2} \)
13 \( 1 + 0.125T + 2.19e3T^{2} \)
17 \( 1 - 57.2T + 4.91e3T^{2} \)
19 \( 1 - 40.8T + 6.85e3T^{2} \)
23 \( 1 - 190.T + 1.21e4T^{2} \)
29 \( 1 - 133.T + 2.43e4T^{2} \)
31 \( 1 - 129.T + 2.97e4T^{2} \)
37 \( 1 + 364.T + 5.06e4T^{2} \)
41 \( 1 - 195.T + 6.89e4T^{2} \)
43 \( 1 - 31.6T + 7.95e4T^{2} \)
47 \( 1 - 479.T + 1.03e5T^{2} \)
53 \( 1 + 402.T + 1.48e5T^{2} \)
61 \( 1 + 209.T + 2.26e5T^{2} \)
67 \( 1 + 455.T + 3.00e5T^{2} \)
71 \( 1 - 203.T + 3.57e5T^{2} \)
73 \( 1 + 177.T + 3.89e5T^{2} \)
79 \( 1 + 491.T + 4.93e5T^{2} \)
83 \( 1 + 717.T + 5.71e5T^{2} \)
89 \( 1 - 1.30e3T + 7.04e5T^{2} \)
97 \( 1 - 538.T + 9.12e5T^{2} \)
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

−12.52612082067777933265356103624, −10.73263478747219918685370943963, −10.18086675865316570931491978584, −9.168833582192823555972593894760, −8.262542199257191246152222977619, −7.45992878832344275290224578635, −5.44446467169540000369207772814, −4.86128461949867782747229504820, −2.75514719633032800620745044377, −1.29915326717631840793766491340, 1.29915326717631840793766491340, 2.75514719633032800620745044377, 4.86128461949867782747229504820, 5.44446467169540000369207772814, 7.45992878832344275290224578635, 8.262542199257191246152222977619, 9.168833582192823555972593894760, 10.18086675865316570931491978584, 10.73263478747219918685370943963, 12.52612082067777933265356103624

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