Properties

Label 2-177-1.1-c3-0-27
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  + 1.29·2-s + 3·3-s − 6.31·4-s + 7.05·5-s + 3.89·6-s − 33.4·7-s − 18.5·8-s + 9·9-s + 9.15·10-s − 27.5·11-s − 18.9·12-s − 55.6·13-s − 43.4·14-s + 21.1·15-s + 26.3·16-s + 108.·17-s + 11.6·18-s − 141.·19-s − 44.5·20-s − 100.·21-s − 35.8·22-s + 142.·23-s − 55.7·24-s − 75.2·25-s − 72.2·26-s + 27·27-s + 211.·28-s + ⋯
L(s)  = 1  + 0.458·2-s + 0.577·3-s − 0.789·4-s + 0.631·5-s + 0.264·6-s − 1.80·7-s − 0.821·8-s + 0.333·9-s + 0.289·10-s − 0.756·11-s − 0.455·12-s − 1.18·13-s − 0.828·14-s + 0.364·15-s + 0.412·16-s + 1.54·17-s + 0.152·18-s − 1.70·19-s − 0.498·20-s − 1.04·21-s − 0.347·22-s + 1.29·23-s − 0.474·24-s − 0.601·25-s − 0.545·26-s + 0.192·27-s + 1.42·28-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 \( 1 - 1.29T + 8T^{2} \)
5 \( 1 - 7.05T + 125T^{2} \)
7 \( 1 + 33.4T + 343T^{2} \)
11 \( 1 + 27.5T + 1.33e3T^{2} \)
13 \( 1 + 55.6T + 2.19e3T^{2} \)
17 \( 1 - 108.T + 4.91e3T^{2} \)
19 \( 1 + 141.T + 6.85e3T^{2} \)
23 \( 1 - 142.T + 1.21e4T^{2} \)
29 \( 1 + 97.2T + 2.43e4T^{2} \)
31 \( 1 + 221.T + 2.97e4T^{2} \)
37 \( 1 - 339.T + 5.06e4T^{2} \)
41 \( 1 + 266.T + 6.89e4T^{2} \)
43 \( 1 + 67.2T + 7.95e4T^{2} \)
47 \( 1 + 262.T + 1.03e5T^{2} \)
53 \( 1 - 380.T + 1.48e5T^{2} \)
61 \( 1 + 15.4T + 2.26e5T^{2} \)
67 \( 1 + 172.T + 3.00e5T^{2} \)
71 \( 1 + 616.T + 3.57e5T^{2} \)
73 \( 1 + 210.T + 3.89e5T^{2} \)
79 \( 1 - 543.T + 4.93e5T^{2} \)
83 \( 1 - 350.T + 5.71e5T^{2} \)
89 \( 1 - 1.44e3T + 7.04e5T^{2} \)
97 \( 1 + 625.T + 9.12e5T^{2} \)
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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

−12.31201052886129989050763436039, −10.27882062286846095309248542703, −9.712266695242400467227604220773, −8.968091682787389481530675152872, −7.55687599673244993489344278411, −6.23391131525252373794030154958, −5.19269740561505549037638280512, −3.67390495373758240107983465066, −2.65515941951464657731438894273, 0, 2.65515941951464657731438894273, 3.67390495373758240107983465066, 5.19269740561505549037638280512, 6.23391131525252373794030154958, 7.55687599673244993489344278411, 8.968091682787389481530675152872, 9.712266695242400467227604220773, 10.27882062286846095309248542703, 12.31201052886129989050763436039

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