Properties

Label 2-177-1.1-c9-0-58
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  − 39.7·2-s − 81·3-s + 1.06e3·4-s + 1.15e3·5-s + 3.22e3·6-s − 1.75e3·7-s − 2.21e4·8-s + 6.56e3·9-s − 4.58e4·10-s + 7.41e4·11-s − 8.66e4·12-s + 1.24e5·13-s + 6.97e4·14-s − 9.33e4·15-s + 3.34e5·16-s − 2.82e5·17-s − 2.60e5·18-s + 4.98e5·19-s + 1.23e6·20-s + 1.42e5·21-s − 2.94e6·22-s − 1.32e6·23-s + 1.79e6·24-s − 6.24e5·25-s − 4.95e6·26-s − 5.31e5·27-s − 1.87e6·28-s + ⋯
L(s)  = 1  − 1.75·2-s − 0.577·3-s + 2.08·4-s + 0.824·5-s + 1.01·6-s − 0.276·7-s − 1.91·8-s + 0.333·9-s − 1.44·10-s + 1.52·11-s − 1.20·12-s + 1.20·13-s + 0.485·14-s − 0.476·15-s + 1.27·16-s − 0.821·17-s − 0.585·18-s + 0.876·19-s + 1.72·20-s + 0.159·21-s − 2.68·22-s − 0.986·23-s + 1.10·24-s − 0.319·25-s − 2.12·26-s − 0.192·27-s − 0.576·28-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 \( 1 + 39.7T + 512T^{2} \)
5 \( 1 - 1.15e3T + 1.95e6T^{2} \)
7 \( 1 + 1.75e3T + 4.03e7T^{2} \)
11 \( 1 - 7.41e4T + 2.35e9T^{2} \)
13 \( 1 - 1.24e5T + 1.06e10T^{2} \)
17 \( 1 + 2.82e5T + 1.18e11T^{2} \)
19 \( 1 - 4.98e5T + 3.22e11T^{2} \)
23 \( 1 + 1.32e6T + 1.80e12T^{2} \)
29 \( 1 + 2.58e6T + 1.45e13T^{2} \)
31 \( 1 + 8.69e6T + 2.64e13T^{2} \)
37 \( 1 + 1.67e7T + 1.29e14T^{2} \)
41 \( 1 - 3.96e6T + 3.27e14T^{2} \)
43 \( 1 - 3.80e7T + 5.02e14T^{2} \)
47 \( 1 - 8.57e6T + 1.11e15T^{2} \)
53 \( 1 + 2.60e7T + 3.29e15T^{2} \)
61 \( 1 - 1.24e8T + 1.16e16T^{2} \)
67 \( 1 + 2.73e8T + 2.72e16T^{2} \)
71 \( 1 - 9.64e7T + 4.58e16T^{2} \)
73 \( 1 + 4.37e8T + 5.88e16T^{2} \)
79 \( 1 + 3.71e8T + 1.19e17T^{2} \)
83 \( 1 - 7.07e8T + 1.86e17T^{2} \)
89 \( 1 - 8.00e8T + 3.50e17T^{2} \)
97 \( 1 - 8.70e8T + 7.60e17T^{2} \)
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

−10.35303695024934011761050835262, −9.301152468801901921891605011107, −8.966506420646242096177555965877, −7.49575840630741266464567660508, −6.49345835491205152629913530514, −5.83192343992460609720665366148, −3.76779436165368033656896667602, −1.91576564241285239245506706839, −1.25418767263186712554002108124, 0, 1.25418767263186712554002108124, 1.91576564241285239245506706839, 3.76779436165368033656896667602, 5.83192343992460609720665366148, 6.49345835491205152629913530514, 7.49575840630741266464567660508, 8.966506420646242096177555965877, 9.301152468801901921891605011107, 10.35303695024934011761050835262

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