Properties

Label 2-2169-1.1-c1-0-48
Degree $2$
Conductor $2169$
Sign $-1$
Analytic cond. $17.3195$
Root an. cond. $4.16167$
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  − 1.54·2-s + 0.398·4-s − 0.334·5-s − 4.24·7-s + 2.47·8-s + 0.518·10-s − 0.915·11-s + 4.81·13-s + 6.57·14-s − 4.63·16-s + 5.38·17-s − 4.34·19-s − 0.133·20-s + 1.41·22-s − 8.10·23-s − 4.88·25-s − 7.45·26-s − 1.69·28-s + 6.45·29-s + 10.7·31-s + 2.22·32-s − 8.34·34-s + 1.42·35-s + 5.16·37-s + 6.73·38-s − 0.830·40-s + 0.612·41-s + ⋯
L(s)  = 1  − 1.09·2-s + 0.199·4-s − 0.149·5-s − 1.60·7-s + 0.876·8-s + 0.164·10-s − 0.276·11-s + 1.33·13-s + 1.75·14-s − 1.15·16-s + 1.30·17-s − 0.997·19-s − 0.0298·20-s + 0.302·22-s − 1.69·23-s − 0.977·25-s − 1.46·26-s − 0.319·28-s + 1.19·29-s + 1.93·31-s + 0.393·32-s − 1.43·34-s + 0.240·35-s + 0.849·37-s + 1.09·38-s − 0.131·40-s + 0.0956·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2169\)    =    \(3^{2} \cdot 241\)
Sign: $-1$
Analytic conductor: \(17.3195\)
Root analytic conductor: \(4.16167\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2169,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
241 \( 1 - T \)
good2 \( 1 + 1.54T + 2T^{2} \)
5 \( 1 + 0.334T + 5T^{2} \)
7 \( 1 + 4.24T + 7T^{2} \)
11 \( 1 + 0.915T + 11T^{2} \)
13 \( 1 - 4.81T + 13T^{2} \)
17 \( 1 - 5.38T + 17T^{2} \)
19 \( 1 + 4.34T + 19T^{2} \)
23 \( 1 + 8.10T + 23T^{2} \)
29 \( 1 - 6.45T + 29T^{2} \)
31 \( 1 - 10.7T + 31T^{2} \)
37 \( 1 - 5.16T + 37T^{2} \)
41 \( 1 - 0.612T + 41T^{2} \)
43 \( 1 + 1.85T + 43T^{2} \)
47 \( 1 - 2.21T + 47T^{2} \)
53 \( 1 + 0.00846T + 53T^{2} \)
59 \( 1 + 8.85T + 59T^{2} \)
61 \( 1 - 3.78T + 61T^{2} \)
67 \( 1 - 4.67T + 67T^{2} \)
71 \( 1 + 1.48T + 71T^{2} \)
73 \( 1 + 10.3T + 73T^{2} \)
79 \( 1 + 17.6T + 79T^{2} \)
83 \( 1 + 7.45T + 83T^{2} \)
89 \( 1 - 0.520T + 89T^{2} \)
97 \( 1 + 6.33T + 97T^{2} \)
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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

−8.518730259851255371145912088817, −8.238677721515726289823926023467, −7.34160535147939316760343433975, −6.24109767550795764150821567092, −6.00328357204186478554765697071, −4.42040481849958117264709650213, −3.69135537695197726923674999683, −2.63502179988512222898405046868, −1.18747882564595536654529453565, 0, 1.18747882564595536654529453565, 2.63502179988512222898405046868, 3.69135537695197726923674999683, 4.42040481849958117264709650213, 6.00328357204186478554765697071, 6.24109767550795764150821567092, 7.34160535147939316760343433975, 8.238677721515726289823926023467, 8.518730259851255371145912088817

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