Properties

Label 2-2166-1.1-c1-0-41
Degree $2$
Conductor $2166$
Sign $-1$
Analytic cond. $17.2955$
Root an. cond. $4.15879$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 1.61·5-s − 6-s − 8-s + 9-s + 1.61·10-s − 4·11-s + 12-s − 0.618·13-s − 1.61·15-s + 16-s + 7.09·17-s − 18-s − 1.61·20-s + 4·22-s − 4·23-s − 24-s − 2.38·25-s + 0.618·26-s + 27-s − 2.14·29-s + 1.61·30-s + 10.4·31-s − 32-s − 4·33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.723·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.511·10-s − 1.20·11-s + 0.288·12-s − 0.171·13-s − 0.417·15-s + 0.250·16-s + 1.71·17-s − 0.235·18-s − 0.361·20-s + 0.852·22-s − 0.834·23-s − 0.204·24-s − 0.476·25-s + 0.121·26-s + 0.192·27-s − 0.398·29-s + 0.295·30-s + 1.88·31-s − 0.176·32-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2166\)    =    \(2 \cdot 3 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(17.2955\)
Root analytic conductor: \(4.15879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2166,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
19 \( 1 \)
good5 \( 1 + 1.61T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 0.618T + 13T^{2} \)
17 \( 1 - 7.09T + 17T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 2.14T + 29T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 + 0.854T + 37T^{2} \)
41 \( 1 + 0.854T + 41T^{2} \)
43 \( 1 + 1.52T + 43T^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 + 6.38T + 53T^{2} \)
59 \( 1 + 6.47T + 59T^{2} \)
61 \( 1 + 7.38T + 61T^{2} \)
67 \( 1 - 15.4T + 67T^{2} \)
71 \( 1 + 2.47T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−8.319919433686698421838341544832, −8.024977885173213731235973985050, −7.58423129413612126555166449990, −6.53844986343870282741077518178, −5.55221809243616450887418830939, −4.58247138454900292960903380328, −3.43602963638392449961245525625, −2.78222236276994469773121213760, −1.52329535296591515958877936932, 0, 1.52329535296591515958877936932, 2.78222236276994469773121213760, 3.43602963638392449961245525625, 4.58247138454900292960903380328, 5.55221809243616450887418830939, 6.53844986343870282741077518178, 7.58423129413612126555166449990, 8.024977885173213731235973985050, 8.319919433686698421838341544832

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