Properties

Label 2-177-1.1-c3-0-15
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.58·2-s + 3·3-s + 12.9·4-s − 12.2·5-s − 13.7·6-s + 15.3·7-s − 22.8·8-s + 9·9-s + 56.3·10-s − 41.7·11-s + 38.9·12-s − 23.4·13-s − 70.3·14-s − 36.8·15-s + 0.870·16-s + 123.·17-s − 41.2·18-s + 106.·19-s − 159.·20-s + 46.0·21-s + 191.·22-s − 207.·23-s − 68.6·24-s + 26.2·25-s + 107.·26-s + 27·27-s + 199.·28-s + ⋯
L(s)  = 1  − 1.61·2-s + 0.577·3-s + 1.62·4-s − 1.09·5-s − 0.935·6-s + 0.828·7-s − 1.01·8-s + 0.333·9-s + 1.78·10-s − 1.14·11-s + 0.937·12-s − 0.501·13-s − 1.34·14-s − 0.634·15-s + 0.0135·16-s + 1.76·17-s − 0.539·18-s + 1.28·19-s − 1.78·20-s + 0.478·21-s + 1.85·22-s − 1.87·23-s − 0.583·24-s + 0.209·25-s + 0.812·26-s + 0.192·27-s + 1.34·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ -1)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
59 \( 1 + 59T \)
good2 \( 1 + 4.58T + 8T^{2} \)
5 \( 1 + 12.2T + 125T^{2} \)
7 \( 1 - 15.3T + 343T^{2} \)
11 \( 1 + 41.7T + 1.33e3T^{2} \)
13 \( 1 + 23.4T + 2.19e3T^{2} \)
17 \( 1 - 123.T + 4.91e3T^{2} \)
19 \( 1 - 106.T + 6.85e3T^{2} \)
23 \( 1 + 207.T + 1.21e4T^{2} \)
29 \( 1 + 53.4T + 2.43e4T^{2} \)
31 \( 1 + 252.T + 2.97e4T^{2} \)
37 \( 1 + 357.T + 5.06e4T^{2} \)
41 \( 1 + 353.T + 6.89e4T^{2} \)
43 \( 1 + 80.6T + 7.95e4T^{2} \)
47 \( 1 + 5.12T + 1.03e5T^{2} \)
53 \( 1 + 260.T + 1.48e5T^{2} \)
61 \( 1 - 35.7T + 2.26e5T^{2} \)
67 \( 1 - 635.T + 3.00e5T^{2} \)
71 \( 1 + 644.T + 3.57e5T^{2} \)
73 \( 1 + 531.T + 3.89e5T^{2} \)
79 \( 1 - 594.T + 4.93e5T^{2} \)
83 \( 1 + 95.7T + 5.71e5T^{2} \)
89 \( 1 - 1.00e3T + 7.04e5T^{2} \)
97 \( 1 + 964.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.56145825081589440025304004999, −10.40431384428572981715729801812, −9.729110876850934454369251929038, −8.412364842605311150912589788182, −7.74284295532676236411256600617, −7.44892986149998045460506035678, −5.23310383944820812807097873916, −3.44316058969392398437979652052, −1.74897375990281298245142082470, 0, 1.74897375990281298245142082470, 3.44316058969392398437979652052, 5.23310383944820812807097873916, 7.44892986149998045460506035678, 7.74284295532676236411256600617, 8.412364842605311150912589788182, 9.729110876850934454369251929038, 10.40431384428572981715729801812, 11.56145825081589440025304004999

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