Properties

Label 2-177-1.1-c9-0-62
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  + 2.31·2-s + 81·3-s − 506.·4-s − 66.4·5-s + 187.·6-s − 1.90e3·7-s − 2.35e3·8-s + 6.56e3·9-s − 153.·10-s − 5.12e4·11-s − 4.10e4·12-s + 1.18e5·13-s − 4.40e3·14-s − 5.38e3·15-s + 2.53e5·16-s + 4.12e5·17-s + 1.51e4·18-s + 2.16e5·19-s + 3.36e4·20-s − 1.54e5·21-s − 1.18e5·22-s − 7.30e5·23-s − 1.90e5·24-s − 1.94e6·25-s + 2.73e5·26-s + 5.31e5·27-s + 9.65e5·28-s + ⋯
L(s)  = 1  + 0.102·2-s + 0.577·3-s − 0.989·4-s − 0.0475·5-s + 0.0589·6-s − 0.299·7-s − 0.203·8-s + 0.333·9-s − 0.00485·10-s − 1.05·11-s − 0.571·12-s + 1.14·13-s − 0.0306·14-s − 0.0274·15-s + 0.968·16-s + 1.19·17-s + 0.0340·18-s + 0.380·19-s + 0.0470·20-s − 0.173·21-s − 0.107·22-s − 0.544·23-s − 0.117·24-s − 0.997·25-s + 0.117·26-s + 0.192·27-s + 0.296·28-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 \( 1 - 2.31T + 512T^{2} \)
5 \( 1 + 66.4T + 1.95e6T^{2} \)
7 \( 1 + 1.90e3T + 4.03e7T^{2} \)
11 \( 1 + 5.12e4T + 2.35e9T^{2} \)
13 \( 1 - 1.18e5T + 1.06e10T^{2} \)
17 \( 1 - 4.12e5T + 1.18e11T^{2} \)
19 \( 1 - 2.16e5T + 3.22e11T^{2} \)
23 \( 1 + 7.30e5T + 1.80e12T^{2} \)
29 \( 1 + 2.16e6T + 1.45e13T^{2} \)
31 \( 1 - 7.53e6T + 2.64e13T^{2} \)
37 \( 1 - 2.06e7T + 1.29e14T^{2} \)
41 \( 1 + 3.10e7T + 3.27e14T^{2} \)
43 \( 1 + 1.98e7T + 5.02e14T^{2} \)
47 \( 1 + 2.28e7T + 1.11e15T^{2} \)
53 \( 1 + 3.27e7T + 3.29e15T^{2} \)
61 \( 1 + 8.93e7T + 1.16e16T^{2} \)
67 \( 1 - 7.39e7T + 2.72e16T^{2} \)
71 \( 1 - 3.75e8T + 4.58e16T^{2} \)
73 \( 1 + 3.48e7T + 5.88e16T^{2} \)
79 \( 1 + 4.51e8T + 1.19e17T^{2} \)
83 \( 1 + 4.70e8T + 1.86e17T^{2} \)
89 \( 1 - 2.05e8T + 3.50e17T^{2} \)
97 \( 1 + 9.28e8T + 7.60e17T^{2} \)
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

−10.12834837060825050783333427588, −9.637309365171516073591968398410, −8.293268022931550338903607992324, −7.890378667947738098197931796867, −6.16792560563194009870898511071, −5.10611581303439097068433776700, −3.85490103653558658591812958338, −2.99757071276347937436583310151, −1.31701410908939947319103914137, 0, 1.31701410908939947319103914137, 2.99757071276347937436583310151, 3.85490103653558658591812958338, 5.10611581303439097068433776700, 6.16792560563194009870898511071, 7.890378667947738098197931796867, 8.293268022931550338903607992324, 9.637309365171516073591968398410, 10.12834837060825050783333427588

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