Properties

Label 2-2166-1.1-c1-0-31
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 − 4·5-s − 6-s − 3·7-s − 8-s + 9-s + 4·10-s + 2·11-s + 12-s + 7·13-s + 3·14-s − 4·15-s + 16-s − 18-s − 4·20-s − 3·21-s − 2·22-s − 4·23-s − 24-s + 11·25-s − 7·26-s + 27-s − 3·28-s − 4·29-s + 4·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.603·11-s + 0.288·12-s + 1.94·13-s + 0.801·14-s − 1.03·15-s + 1/4·16-s − 0.235·18-s − 0.894·20-s − 0.654·21-s − 0.426·22-s − 0.834·23-s − 0.204·24-s + 11/5·25-s − 1.37·26-s + 0.192·27-s − 0.566·28-s − 0.742·29-s + 0.730·30-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 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−8.632336773486661306730920493015, −8.119209499510440553100289543051, −7.23066919257894410409371315840, −6.68461495565388836489640760645, −5.76509452700412448313739003075, −4.00716186536001279521923234189, −3.78769641466580568623112365695, −2.95181659001229895680726890061, −1.32962279936782404079285849890, 0, 1.32962279936782404079285849890, 2.95181659001229895680726890061, 3.78769641466580568623112365695, 4.00716186536001279521923234189, 5.76509452700412448313739003075, 6.68461495565388836489640760645, 7.23066919257894410409371315840, 8.119209499510440553100289543051, 8.632336773486661306730920493015

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