Properties

Label 2-2166-1.1-c1-0-12
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 − 3.41·5-s + 6-s − 2.87·7-s + 8-s + 9-s − 3.41·10-s − 0.347·11-s + 12-s + 1.65·13-s − 2.87·14-s − 3.41·15-s + 16-s + 6.94·17-s + 18-s − 3.41·20-s − 2.87·21-s − 0.347·22-s + 6.80·23-s + 24-s + 6.63·25-s + 1.65·26-s + 27-s − 2.87·28-s − 6.35·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.52·5-s + 0.408·6-s − 1.08·7-s + 0.353·8-s + 0.333·9-s − 1.07·10-s − 0.104·11-s + 0.288·12-s + 0.458·13-s − 0.769·14-s − 0.880·15-s + 0.250·16-s + 1.68·17-s + 0.235·18-s − 0.762·20-s − 0.628·21-s − 0.0740·22-s + 1.41·23-s + 0.204·24-s + 1.32·25-s + 0.324·26-s + 0.192·27-s − 0.544·28-s − 1.18·29-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

\(L(1)\) \(\approx\) \(2.411187186\)
\(L(\frac12)\) \(\approx\) \(2.411187186\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.41T + 5T^{2} \)
7 \( 1 + 2.87T + 7T^{2} \)
11 \( 1 + 0.347T + 11T^{2} \)
13 \( 1 - 1.65T + 13T^{2} \)
17 \( 1 - 6.94T + 17T^{2} \)
23 \( 1 - 6.80T + 23T^{2} \)
29 \( 1 + 6.35T + 29T^{2} \)
31 \( 1 + 1.59T + 31T^{2} \)
37 \( 1 - 11.2T + 37T^{2} \)
41 \( 1 - 3.49T + 41T^{2} \)
43 \( 1 - 2.28T + 43T^{2} \)
47 \( 1 - 5.59T + 47T^{2} \)
53 \( 1 + 1.98T + 53T^{2} \)
59 \( 1 + 0.445T + 59T^{2} \)
61 \( 1 + 12.5T + 61T^{2} \)
67 \( 1 - 1.07T + 67T^{2} \)
71 \( 1 - 16.6T + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + 1.79T + 89T^{2} \)
97 \( 1 + 3.65T + 97T^{2} \)
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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

−9.126283023430689494607082603139, −7.921316383429760413081432946983, −7.66855515502088286048426249841, −6.81501965247708818078862204191, −5.91563474198050230805927988420, −4.91400769963306670568693244064, −3.82244159939346923760838703742, −3.49705626998349679384245393045, −2.70141910927224720419609126593, −0.904353645007629231783193235115, 0.904353645007629231783193235115, 2.70141910927224720419609126593, 3.49705626998349679384245393045, 3.82244159939346923760838703742, 4.91400769963306670568693244064, 5.91563474198050230805927988420, 6.81501965247708818078862204191, 7.66855515502088286048426249841, 7.921316383429760413081432946983, 9.126283023430689494607082603139

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