Properties

Label 2-177-1.1-c3-0-11
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  − 2.61·2-s − 3·3-s − 1.14·4-s − 7.96·5-s + 7.85·6-s + 16.8·7-s + 23.9·8-s + 9·9-s + 20.8·10-s + 62.7·11-s + 3.42·12-s − 75.2·13-s − 44.1·14-s + 23.9·15-s − 53.5·16-s − 51.9·17-s − 23.5·18-s + 24.2·19-s + 9.09·20-s − 50.6·21-s − 164.·22-s − 9.85·23-s − 71.8·24-s − 61.5·25-s + 196.·26-s − 27·27-s − 19.2·28-s + ⋯
L(s)  = 1  − 0.925·2-s − 0.577·3-s − 0.142·4-s − 0.712·5-s + 0.534·6-s + 0.911·7-s + 1.05·8-s + 0.333·9-s + 0.659·10-s + 1.72·11-s + 0.0823·12-s − 1.60·13-s − 0.843·14-s + 0.411·15-s − 0.836·16-s − 0.741·17-s − 0.308·18-s + 0.292·19-s + 0.101·20-s − 0.526·21-s − 1.59·22-s − 0.0893·23-s − 0.610·24-s − 0.492·25-s + 1.48·26-s − 0.192·27-s − 0.129·28-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 \( 1 + 2.61T + 8T^{2} \)
5 \( 1 + 7.96T + 125T^{2} \)
7 \( 1 - 16.8T + 343T^{2} \)
11 \( 1 - 62.7T + 1.33e3T^{2} \)
13 \( 1 + 75.2T + 2.19e3T^{2} \)
17 \( 1 + 51.9T + 4.91e3T^{2} \)
19 \( 1 - 24.2T + 6.85e3T^{2} \)
23 \( 1 + 9.85T + 1.21e4T^{2} \)
29 \( 1 - 160.T + 2.43e4T^{2} \)
31 \( 1 + 125.T + 2.97e4T^{2} \)
37 \( 1 + 224.T + 5.06e4T^{2} \)
41 \( 1 + 467.T + 6.89e4T^{2} \)
43 \( 1 - 90.9T + 7.95e4T^{2} \)
47 \( 1 - 437.T + 1.03e5T^{2} \)
53 \( 1 + 353.T + 1.48e5T^{2} \)
61 \( 1 + 697.T + 2.26e5T^{2} \)
67 \( 1 + 292.T + 3.00e5T^{2} \)
71 \( 1 - 223.T + 3.57e5T^{2} \)
73 \( 1 - 122.T + 3.89e5T^{2} \)
79 \( 1 + 440.T + 4.93e5T^{2} \)
83 \( 1 + 662.T + 5.71e5T^{2} \)
89 \( 1 - 1.01e3T + 7.04e5T^{2} \)
97 \( 1 + 397.T + 9.12e5T^{2} \)
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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

−11.71161811450465143587196369629, −10.71144461773026984886634169203, −9.631672300621368549961105600445, −8.744507303002251351049037388776, −7.67095305676505166620895010483, −6.82151572149032375375175816024, −4.98330391686472106529166112028, −4.13406633749339021797690890030, −1.57054288843129401275024617155, 0, 1.57054288843129401275024617155, 4.13406633749339021797690890030, 4.98330391686472106529166112028, 6.82151572149032375375175816024, 7.67095305676505166620895010483, 8.744507303002251351049037388776, 9.631672300621368549961105600445, 10.71144461773026984886634169203, 11.71161811450465143587196369629

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