Properties

Label 2-177-1.1-c13-0-34
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $189.798$
Root an. cond. $13.7767$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 149.·2-s − 729·3-s + 1.41e4·4-s − 4.87e4·5-s + 1.08e5·6-s − 3.00e5·7-s − 8.84e5·8-s + 5.31e5·9-s + 7.27e6·10-s + 2.91e6·11-s − 1.02e7·12-s + 4.39e6·13-s + 4.48e7·14-s + 3.55e7·15-s + 1.65e7·16-s − 6.68e7·17-s − 7.93e7·18-s − 3.05e8·19-s − 6.87e8·20-s + 2.18e8·21-s − 4.35e8·22-s − 4.79e8·23-s + 6.45e8·24-s + 1.15e9·25-s − 6.55e8·26-s − 3.87e8·27-s − 4.23e9·28-s + ⋯
L(s)  = 1  − 1.65·2-s − 0.577·3-s + 1.72·4-s − 1.39·5-s + 0.952·6-s − 0.964·7-s − 1.19·8-s + 0.333·9-s + 2.30·10-s + 0.496·11-s − 0.994·12-s + 0.252·13-s + 1.59·14-s + 0.804·15-s + 0.246·16-s − 0.671·17-s − 0.550·18-s − 1.48·19-s − 2.40·20-s + 0.556·21-s − 0.819·22-s − 0.674·23-s + 0.689·24-s + 0.943·25-s − 0.416·26-s − 0.192·27-s − 1.66·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(189.798\)
Root analytic conductor: \(13.7767\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 729T \)
59 \( 1 + 4.21e10T \)
good2 \( 1 + 149.T + 8.19e3T^{2} \)
5 \( 1 + 4.87e4T + 1.22e9T^{2} \)
7 \( 1 + 3.00e5T + 9.68e10T^{2} \)
11 \( 1 - 2.91e6T + 3.45e13T^{2} \)
13 \( 1 - 4.39e6T + 3.02e14T^{2} \)
17 \( 1 + 6.68e7T + 9.90e15T^{2} \)
19 \( 1 + 3.05e8T + 4.20e16T^{2} \)
23 \( 1 + 4.79e8T + 5.04e17T^{2} \)
29 \( 1 - 1.34e9T + 1.02e19T^{2} \)
31 \( 1 - 8.39e9T + 2.44e19T^{2} \)
37 \( 1 + 2.42e10T + 2.43e20T^{2} \)
41 \( 1 + 2.76e10T + 9.25e20T^{2} \)
43 \( 1 + 7.37e10T + 1.71e21T^{2} \)
47 \( 1 + 5.77e10T + 5.46e21T^{2} \)
53 \( 1 - 1.52e11T + 2.60e22T^{2} \)
61 \( 1 - 6.51e11T + 1.61e23T^{2} \)
67 \( 1 + 1.13e12T + 5.48e23T^{2} \)
71 \( 1 + 9.22e10T + 1.16e24T^{2} \)
73 \( 1 - 2.04e12T + 1.67e24T^{2} \)
79 \( 1 - 1.06e12T + 4.66e24T^{2} \)
83 \( 1 - 1.85e12T + 8.87e24T^{2} \)
89 \( 1 - 2.98e12T + 2.19e25T^{2} \)
97 \( 1 + 1.02e13T + 6.73e25T^{2} \)
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

−9.924143613691943836487760009377, −8.663450750141954788436288415682, −8.167502000178221833638653973208, −6.80904054212156258831695361970, −6.51347147169880175337993453319, −4.51430367978566215612536697291, −3.42661186457331251005089389539, −1.93032719649648150665125364841, −0.60677253722111931302609589053, 0, 0.60677253722111931302609589053, 1.93032719649648150665125364841, 3.42661186457331251005089389539, 4.51430367978566215612536697291, 6.51347147169880175337993453319, 6.80904054212156258831695361970, 8.167502000178221833638653973208, 8.663450750141954788436288415682, 9.924143613691943836487760009377

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