Properties

Label 2-177-1.1-c9-0-80
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 29.8·2-s − 81·3-s + 376.·4-s + 2.38e3·5-s − 2.41e3·6-s + 4.53e3·7-s − 4.04e3·8-s + 6.56e3·9-s + 7.11e4·10-s − 3.86e4·11-s − 3.04e4·12-s − 1.59e5·13-s + 1.35e5·14-s − 1.93e5·15-s − 3.13e5·16-s − 2.83e5·17-s + 1.95e5·18-s − 4.54e5·19-s + 8.97e5·20-s − 3.67e5·21-s − 1.15e6·22-s − 2.42e6·23-s + 3.27e5·24-s + 3.74e6·25-s − 4.75e6·26-s − 5.31e5·27-s + 1.70e6·28-s + ⋯
L(s)  = 1  + 1.31·2-s − 0.577·3-s + 0.734·4-s + 1.70·5-s − 0.760·6-s + 0.714·7-s − 0.349·8-s + 0.333·9-s + 2.24·10-s − 0.795·11-s − 0.424·12-s − 1.54·13-s + 0.940·14-s − 0.985·15-s − 1.19·16-s − 0.822·17-s + 0.439·18-s − 0.799·19-s + 1.25·20-s − 0.412·21-s − 1.04·22-s − 1.80·23-s + 0.201·24-s + 1.91·25-s − 2.04·26-s − 0.192·27-s + 0.524·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 81T \)
59 \( 1 + 1.21e7T \)
good2 \( 1 - 29.8T + 512T^{2} \)
5 \( 1 - 2.38e3T + 1.95e6T^{2} \)
7 \( 1 - 4.53e3T + 4.03e7T^{2} \)
11 \( 1 + 3.86e4T + 2.35e9T^{2} \)
13 \( 1 + 1.59e5T + 1.06e10T^{2} \)
17 \( 1 + 2.83e5T + 1.18e11T^{2} \)
19 \( 1 + 4.54e5T + 3.22e11T^{2} \)
23 \( 1 + 2.42e6T + 1.80e12T^{2} \)
29 \( 1 - 6.69e6T + 1.45e13T^{2} \)
31 \( 1 - 4.53e6T + 2.64e13T^{2} \)
37 \( 1 + 1.75e7T + 1.29e14T^{2} \)
41 \( 1 + 1.25e7T + 3.27e14T^{2} \)
43 \( 1 - 2.32e4T + 5.02e14T^{2} \)
47 \( 1 + 5.96e6T + 1.11e15T^{2} \)
53 \( 1 - 8.21e7T + 3.29e15T^{2} \)
61 \( 1 + 6.81e7T + 1.16e16T^{2} \)
67 \( 1 + 6.03e7T + 2.72e16T^{2} \)
71 \( 1 + 7.46e7T + 4.58e16T^{2} \)
73 \( 1 - 1.87e8T + 5.88e16T^{2} \)
79 \( 1 + 1.18e8T + 1.19e17T^{2} \)
83 \( 1 + 3.24e8T + 1.86e17T^{2} \)
89 \( 1 + 2.07e7T + 3.50e17T^{2} \)
97 \( 1 + 6.27e8T + 7.60e17T^{2} \)
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

−10.49288101340161521462749292299, −9.901311690162574286596179336901, −8.496831128211771592397674610402, −6.81547705790542375521948349071, −5.98171202992815714619697045087, −5.08376106288854755332498631501, −4.54977955076310513866165211773, −2.59328785862954694501912483754, −1.93017985116448036024111158078, 0, 1.93017985116448036024111158078, 2.59328785862954694501912483754, 4.54977955076310513866165211773, 5.08376106288854755332498631501, 5.98171202992815714619697045087, 6.81547705790542375521948349071, 8.496831128211771592397674610402, 9.901311690162574286596179336901, 10.49288101340161521462749292299

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