Properties

Label 2-2166-1.1-c1-0-29
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 2.22·5-s + 6-s − 0.652·7-s + 8-s + 9-s + 2.22·10-s − 1.53·11-s + 12-s + 0.467·13-s − 0.652·14-s + 2.22·15-s + 16-s − 2.10·17-s + 18-s + 2.22·20-s − 0.652·21-s − 1.53·22-s + 5.90·23-s + 24-s − 0.0418·25-s + 0.467·26-s + 27-s − 0.652·28-s + 8.33·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.995·5-s + 0.408·6-s − 0.246·7-s + 0.353·8-s + 0.333·9-s + 0.704·10-s − 0.461·11-s + 0.288·12-s + 0.129·13-s − 0.174·14-s + 0.574·15-s + 0.250·16-s − 0.510·17-s + 0.235·18-s + 0.497·20-s − 0.142·21-s − 0.326·22-s + 1.23·23-s + 0.204·24-s − 0.00837·25-s + 0.0917·26-s + 0.192·27-s − 0.123·28-s + 1.54·29-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: $\chi_{2166} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2166,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.105103262\)
\(L(\frac12)\) \(\approx\) \(4.105103262\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.22T + 5T^{2} \)
7 \( 1 + 0.652T + 7T^{2} \)
11 \( 1 + 1.53T + 11T^{2} \)
13 \( 1 - 0.467T + 13T^{2} \)
17 \( 1 + 2.10T + 17T^{2} \)
23 \( 1 - 5.90T + 23T^{2} \)
29 \( 1 - 8.33T + 29T^{2} \)
31 \( 1 - 8.63T + 31T^{2} \)
37 \( 1 - 4.67T + 37T^{2} \)
41 \( 1 + 3.47T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + 4.63T + 47T^{2} \)
53 \( 1 - 11.9T + 53T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 + 9.12T + 61T^{2} \)
67 \( 1 + 0.248T + 67T^{2} \)
71 \( 1 - 4.44T + 71T^{2} \)
73 \( 1 + 9.09T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 - 3.70T + 83T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 + 2.46T + 97T^{2} \)
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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

−9.081940700421271539258346796728, −8.333524126661174950034714249865, −7.41933966318204320025041704707, −6.52999800085345835332225270461, −5.99628739186168474857195042971, −4.96909058163643420547256287185, −4.30934508866633252469056170567, −2.98605416111299620352314484815, −2.55502466899488522267943121322, −1.30546881369965573227079593671, 1.30546881369965573227079593671, 2.55502466899488522267943121322, 2.98605416111299620352314484815, 4.30934508866633252469056170567, 4.96909058163643420547256287185, 5.99628739186168474857195042971, 6.52999800085345835332225270461, 7.41933966318204320025041704707, 8.333524126661174950034714249865, 9.081940700421271539258346796728

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