Properties

Label 2-177-1.1-c5-0-18
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
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  − 2.87·2-s − 9·3-s − 23.7·4-s − 77.5·5-s + 25.9·6-s + 45.2·7-s + 160.·8-s + 81·9-s + 223.·10-s + 155.·11-s + 213.·12-s + 523.·13-s − 130.·14-s + 697.·15-s + 296.·16-s + 1.58e3·17-s − 233.·18-s − 2.04e3·19-s + 1.83e3·20-s − 407.·21-s − 447.·22-s − 1.06e3·23-s − 1.44e3·24-s + 2.88e3·25-s − 1.50e3·26-s − 729·27-s − 1.07e3·28-s + ⋯
L(s)  = 1  − 0.509·2-s − 0.577·3-s − 0.740·4-s − 1.38·5-s + 0.293·6-s + 0.349·7-s + 0.886·8-s + 0.333·9-s + 0.706·10-s + 0.387·11-s + 0.427·12-s + 0.858·13-s − 0.177·14-s + 0.800·15-s + 0.289·16-s + 1.33·17-s − 0.169·18-s − 1.29·19-s + 1.02·20-s − 0.201·21-s − 0.197·22-s − 0.419·23-s − 0.511·24-s + 0.923·25-s − 0.437·26-s − 0.192·27-s − 0.258·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9T \)
59 \( 1 + 3.48e3T \)
good2 \( 1 + 2.87T + 32T^{2} \)
5 \( 1 + 77.5T + 3.12e3T^{2} \)
7 \( 1 - 45.2T + 1.68e4T^{2} \)
11 \( 1 - 155.T + 1.61e5T^{2} \)
13 \( 1 - 523.T + 3.71e5T^{2} \)
17 \( 1 - 1.58e3T + 1.41e6T^{2} \)
19 \( 1 + 2.04e3T + 2.47e6T^{2} \)
23 \( 1 + 1.06e3T + 6.43e6T^{2} \)
29 \( 1 - 2.23e3T + 2.05e7T^{2} \)
31 \( 1 + 8.69e3T + 2.86e7T^{2} \)
37 \( 1 - 1.50e4T + 6.93e7T^{2} \)
41 \( 1 + 864.T + 1.15e8T^{2} \)
43 \( 1 - 1.13e4T + 1.47e8T^{2} \)
47 \( 1 - 5.53e3T + 2.29e8T^{2} \)
53 \( 1 - 1.80e4T + 4.18e8T^{2} \)
61 \( 1 + 4.43e4T + 8.44e8T^{2} \)
67 \( 1 + 3.33e4T + 1.35e9T^{2} \)
71 \( 1 - 2.86e3T + 1.80e9T^{2} \)
73 \( 1 + 6.37e4T + 2.07e9T^{2} \)
79 \( 1 - 8.29e4T + 3.07e9T^{2} \)
83 \( 1 + 6.79e4T + 3.93e9T^{2} \)
89 \( 1 - 2.25e4T + 5.58e9T^{2} \)
97 \( 1 - 2.62e4T + 8.58e9T^{2} \)
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

−11.21052745332638084477579369313, −10.45026995263911108459415589616, −9.148459751955065655244412100703, −8.163218670159167827850923581440, −7.48416045198349107857120829384, −5.92399911141461447295101716448, −4.49439100604062719263124589079, −3.74941518126695962832085135375, −1.18000841938274987069524701550, 0, 1.18000841938274987069524701550, 3.74941518126695962832085135375, 4.49439100604062719263124589079, 5.92399911141461447295101716448, 7.48416045198349107857120829384, 8.163218670159167827850923581440, 9.148459751955065655244412100703, 10.45026995263911108459415589616, 11.21052745332638084477579369313

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