Properties

Label 2-177-1.1-c3-0-28
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 − 17.0·5-s − 13.7·6-s − 20.9·7-s + 22.8·8-s + 9·9-s − 77.9·10-s − 48.1·11-s − 38.9·12-s + 29.4·13-s − 96.1·14-s + 51.0·15-s + 0.636·16-s + 108.·17-s + 41.2·18-s − 111.·19-s − 220.·20-s + 62.9·21-s − 220.·22-s − 14.3·23-s − 68.4·24-s + 164.·25-s + 134.·26-s − 27·27-s − 272.·28-s + ⋯
L(s)  = 1  + 1.61·2-s − 0.577·3-s + 1.62·4-s − 1.52·5-s − 0.934·6-s − 1.13·7-s + 1.00·8-s + 0.333·9-s − 2.46·10-s − 1.32·11-s − 0.936·12-s + 0.628·13-s − 1.83·14-s + 0.878·15-s + 0.00994·16-s + 1.54·17-s + 0.539·18-s − 1.34·19-s − 2.46·20-s + 0.654·21-s − 2.13·22-s − 0.130·23-s − 0.581·24-s + 1.31·25-s + 1.01·26-s − 0.192·27-s − 1.83·28-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 \( 1 + 17.0T + 125T^{2} \)
7 \( 1 + 20.9T + 343T^{2} \)
11 \( 1 + 48.1T + 1.33e3T^{2} \)
13 \( 1 - 29.4T + 2.19e3T^{2} \)
17 \( 1 - 108.T + 4.91e3T^{2} \)
19 \( 1 + 111.T + 6.85e3T^{2} \)
23 \( 1 + 14.3T + 1.21e4T^{2} \)
29 \( 1 - 292.T + 2.43e4T^{2} \)
31 \( 1 + 216.T + 2.97e4T^{2} \)
37 \( 1 + 168.T + 5.06e4T^{2} \)
41 \( 1 + 342.T + 6.89e4T^{2} \)
43 \( 1 - 325.T + 7.95e4T^{2} \)
47 \( 1 + 315.T + 1.03e5T^{2} \)
53 \( 1 - 256.T + 1.48e5T^{2} \)
61 \( 1 + 749.T + 2.26e5T^{2} \)
67 \( 1 - 664.T + 3.00e5T^{2} \)
71 \( 1 - 207.T + 3.57e5T^{2} \)
73 \( 1 - 45.3T + 3.89e5T^{2} \)
79 \( 1 + 813.T + 4.93e5T^{2} \)
83 \( 1 - 19.2T + 5.71e5T^{2} \)
89 \( 1 + 636.T + 7.04e5T^{2} \)
97 \( 1 + 235.T + 9.12e5T^{2} \)
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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

−12.24322666218727003694005816079, −11.09597708495705090801282400495, −10.27073586294479759064694156381, −8.324040983380215526919243669838, −7.15745668259941798909254992798, −6.14939661296469118237415473262, −5.05287762622093793736479334131, −3.88879478007538630779664720436, −3.05841182362752964719482326657, 0, 3.05841182362752964719482326657, 3.88879478007538630779664720436, 5.05287762622093793736479334131, 6.14939661296469118237415473262, 7.15745668259941798909254992798, 8.324040983380215526919243669838, 10.27073586294479759064694156381, 11.09597708495705090801282400495, 12.24322666218727003694005816079

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