Properties

Label 2-2166-1.1-c3-0-113
Degree $2$
Conductor $2166$
Sign $-1$
Analytic cond. $127.798$
Root an. cond. $11.3047$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 4·4-s − 6.95·5-s + 6·6-s + 19.9·7-s − 8·8-s + 9·9-s + 13.9·10-s + 20.9·11-s − 12·12-s + 38.8·13-s − 39.9·14-s + 20.8·15-s + 16·16-s + 12·17-s − 18·18-s − 27.8·20-s − 59.8·21-s − 41.9·22-s + 29.0·23-s + 24·24-s − 76.6·25-s − 77.6·26-s − 27·27-s + 79.8·28-s − 10.0·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.622·5-s + 0.408·6-s + 1.07·7-s − 0.353·8-s + 0.333·9-s + 0.439·10-s + 0.574·11-s − 0.288·12-s + 0.828·13-s − 0.761·14-s + 0.359·15-s + 0.250·16-s + 0.171·17-s − 0.235·18-s − 0.311·20-s − 0.622·21-s − 0.406·22-s + 0.263·23-s + 0.204·24-s − 0.613·25-s − 0.585·26-s − 0.192·27-s + 0.538·28-s − 0.0646·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2166\)    =    \(2 \cdot 3 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(127.798\)
Root analytic conductor: \(11.3047\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2166,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
3 \( 1 + 3T \)
19 \( 1 \)
good5 \( 1 + 6.95T + 125T^{2} \)
7 \( 1 - 19.9T + 343T^{2} \)
11 \( 1 - 20.9T + 1.33e3T^{2} \)
13 \( 1 - 38.8T + 2.19e3T^{2} \)
17 \( 1 - 12T + 4.91e3T^{2} \)
23 \( 1 - 29.0T + 1.21e4T^{2} \)
29 \( 1 + 10.0T + 2.43e4T^{2} \)
31 \( 1 + 229.T + 2.97e4T^{2} \)
37 \( 1 + 262.T + 5.06e4T^{2} \)
41 \( 1 + 122.T + 6.89e4T^{2} \)
43 \( 1 + 234.T + 7.95e4T^{2} \)
47 \( 1 - 611.T + 1.03e5T^{2} \)
53 \( 1 - 119.T + 1.48e5T^{2} \)
59 \( 1 + 259.T + 2.05e5T^{2} \)
61 \( 1 - 82.0T + 2.26e5T^{2} \)
67 \( 1 + 578.T + 3.00e5T^{2} \)
71 \( 1 - 638.T + 3.57e5T^{2} \)
73 \( 1 - 112.T + 3.89e5T^{2} \)
79 \( 1 + 957.T + 4.93e5T^{2} \)
83 \( 1 + 780.T + 5.71e5T^{2} \)
89 \( 1 - 813.T + 7.04e5T^{2} \)
97 \( 1 + 1.48e3T + 9.12e5T^{2} \)
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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

−8.417037459864495135404692325071, −7.55282225583898427019191514220, −7.00809975861580610488536495799, −5.99682947162213409566596285081, −5.25146701900769884147394475986, −4.22190024394487924920300570995, −3.44967812926535985009979042009, −1.90753615650609810385059617998, −1.16734769233290641702361502854, 0, 1.16734769233290641702361502854, 1.90753615650609810385059617998, 3.44967812926535985009979042009, 4.22190024394487924920300570995, 5.25146701900769884147394475986, 5.99682947162213409566596285081, 7.00809975861580610488536495799, 7.55282225583898427019191514220, 8.417037459864495135404692325071

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