Properties

Label 2-2166-1.1-c1-0-53
Degree $2$
Conductor $2166$
Sign $-1$
Analytic cond. $17.2955$
Root an. cond. $4.15879$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 2·5-s − 6-s − 8-s + 9-s − 2·10-s − 4·11-s + 12-s − 2·13-s + 2·15-s + 16-s − 6·17-s − 18-s + 2·20-s + 4·22-s − 4·23-s − 24-s − 25-s + 2·26-s + 27-s + 2·29-s − 2·30-s − 4·31-s − 32-s − 4·33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s + 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.447·20-s + 0.852·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.371·29-s − 0.365·30-s − 0.718·31-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: yes
Arithmetic: yes
Character: $\chi_{2166} (1, \cdot )$
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 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−8.691593993434944817407339534944, −8.100143290691371559447654285071, −7.22153922721882802395635766156, −6.54324711643559493568388573046, −5.55712983394880976325928956483, −4.76383709193481805378371975040, −3.46272294158930356206920299963, −2.33039316998666771503145824429, −1.90797026202998636328713780462, 0, 1.90797026202998636328713780462, 2.33039316998666771503145824429, 3.46272294158930356206920299963, 4.76383709193481805378371975040, 5.55712983394880976325928956483, 6.54324711643559493568388573046, 7.22153922721882802395635766156, 8.100143290691371559447654285071, 8.691593993434944817407339534944

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