Properties

Label 2-177-59.58-c4-0-20
Degree $2$
Conductor $177$
Sign $-0.861 + 0.506i$
Analytic cond. $18.2964$
Root an. cond. $4.27743$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 7.77i·2-s − 5.19·3-s − 44.4·4-s + 28.5·5-s + 40.4i·6-s + 72.7·7-s + 221. i·8-s + 27·9-s − 222. i·10-s − 9.33i·11-s + 231.·12-s − 106. i·13-s − 565. i·14-s − 148.·15-s + 1.01e3·16-s + 337.·17-s + ⋯
L(s)  = 1  − 1.94i·2-s − 0.577·3-s − 2.78·4-s + 1.14·5-s + 1.12i·6-s + 1.48·7-s + 3.46i·8-s + 0.333·9-s − 2.22i·10-s − 0.0771i·11-s + 1.60·12-s − 0.631i·13-s − 2.88i·14-s − 0.660·15-s + 3.95·16-s + 1.16·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 + 0.506i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.861 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.861 + 0.506i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ -0.861 + 0.506i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.907911377\)
\(L(\frac12)\) \(\approx\) \(1.907911377\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.19T \)
59 \( 1 + (3.00e3 - 1.76e3i)T \)
good2 \( 1 + 7.77iT - 16T^{2} \)
5 \( 1 - 28.5T + 625T^{2} \)
7 \( 1 - 72.7T + 2.40e3T^{2} \)
11 \( 1 + 9.33iT - 1.46e4T^{2} \)
13 \( 1 + 106. iT - 2.85e4T^{2} \)
17 \( 1 - 337.T + 8.35e4T^{2} \)
19 \( 1 - 546.T + 1.30e5T^{2} \)
23 \( 1 - 39.3iT - 2.79e5T^{2} \)
29 \( 1 + 444.T + 7.07e5T^{2} \)
31 \( 1 - 1.23e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.94e3iT - 1.87e6T^{2} \)
41 \( 1 + 835.T + 2.82e6T^{2} \)
43 \( 1 + 2.24e3iT - 3.41e6T^{2} \)
47 \( 1 + 3.91e3iT - 4.87e6T^{2} \)
53 \( 1 - 3.82e3T + 7.89e6T^{2} \)
61 \( 1 + 2.34e3iT - 1.38e7T^{2} \)
67 \( 1 + 3.65e3iT - 2.01e7T^{2} \)
71 \( 1 + 2.63e3T + 2.54e7T^{2} \)
73 \( 1 + 8.92e3iT - 2.83e7T^{2} \)
79 \( 1 - 1.49e3T + 3.89e7T^{2} \)
83 \( 1 + 3.98e3iT - 4.74e7T^{2} \)
89 \( 1 + 2.53e3iT - 6.27e7T^{2} \)
97 \( 1 - 9.21e3iT - 8.85e7T^{2} \)
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

−11.73475252307553435975175051643, −10.53884787822142810012936214331, −10.13135846163883607139421642127, −9.029021564189274753103943723834, −7.85598637493359175002661575821, −5.41619110362280766445217156569, −5.07559461754689472530090726069, −3.36117936699528465332474316877, −1.84444735687160826785925655680, −1.03272348902712586266513231581, 1.23535238693375160867901574173, 4.34701267984993355761149260506, 5.43979179015749823361768575456, 5.83868442240192861605093292778, 7.23828110009874929831291291404, 7.942307964730152981983095930855, 9.247744562536117049937069097718, 9.938696788259714419523491993875, 11.47679894547406375539886456512, 12.82341214525722939439234766057

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