Properties

Label 2-177-1.1-c3-0-20
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  − 5.60·2-s + 3·3-s + 23.4·4-s + 12.9·5-s − 16.8·6-s − 18.2·7-s − 86.4·8-s + 9·9-s − 72.7·10-s − 71.8·11-s + 70.2·12-s + 31.4·13-s + 102.·14-s + 38.9·15-s + 297.·16-s − 94.8·17-s − 50.4·18-s − 53.7·19-s + 303.·20-s − 54.8·21-s + 402.·22-s − 70.2·23-s − 259.·24-s + 43.3·25-s − 176.·26-s + 27·27-s − 428.·28-s + ⋯
L(s)  = 1  − 1.98·2-s + 0.577·3-s + 2.92·4-s + 1.16·5-s − 1.14·6-s − 0.988·7-s − 3.81·8-s + 0.333·9-s − 2.29·10-s − 1.96·11-s + 1.69·12-s + 0.671·13-s + 1.95·14-s + 0.670·15-s + 4.64·16-s − 1.35·17-s − 0.660·18-s − 0.648·19-s + 3.39·20-s − 0.570·21-s + 3.90·22-s − 0.636·23-s − 2.20·24-s + 0.346·25-s − 1.33·26-s + 0.192·27-s − 2.89·28-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 \( 1 + 5.60T + 8T^{2} \)
5 \( 1 - 12.9T + 125T^{2} \)
7 \( 1 + 18.2T + 343T^{2} \)
11 \( 1 + 71.8T + 1.33e3T^{2} \)
13 \( 1 - 31.4T + 2.19e3T^{2} \)
17 \( 1 + 94.8T + 4.91e3T^{2} \)
19 \( 1 + 53.7T + 6.85e3T^{2} \)
23 \( 1 + 70.2T + 1.21e4T^{2} \)
29 \( 1 - 20.2T + 2.43e4T^{2} \)
31 \( 1 + 5.95T + 2.97e4T^{2} \)
37 \( 1 - 167.T + 5.06e4T^{2} \)
41 \( 1 - 248.T + 6.89e4T^{2} \)
43 \( 1 + 472.T + 7.95e4T^{2} \)
47 \( 1 + 140.T + 1.03e5T^{2} \)
53 \( 1 + 21.3T + 1.48e5T^{2} \)
61 \( 1 - 375.T + 2.26e5T^{2} \)
67 \( 1 + 410.T + 3.00e5T^{2} \)
71 \( 1 - 420.T + 3.57e5T^{2} \)
73 \( 1 + 865.T + 3.89e5T^{2} \)
79 \( 1 + 53.7T + 4.93e5T^{2} \)
83 \( 1 - 39.4T + 5.71e5T^{2} \)
89 \( 1 + 354.T + 7.04e5T^{2} \)
97 \( 1 + 1.41e3T + 9.12e5T^{2} \)
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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

−11.10677280328102982850931961541, −10.25899702626271319053210031752, −9.743886129165784979380739931287, −8.784126385570542667758491947412, −7.981171944413546645303115251282, −6.73248434856157670572974565095, −5.90033753096494836344236706515, −2.84573739504704145327438329099, −2.01057895566647514118964697868, 0, 2.01057895566647514118964697868, 2.84573739504704145327438329099, 5.90033753096494836344236706515, 6.73248434856157670572974565095, 7.981171944413546645303115251282, 8.784126385570542667758491947412, 9.743886129165784979380739931287, 10.25899702626271319053210031752, 11.10677280328102982850931961541

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