Properties

Label 2-2166-1.1-c3-0-157
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 − 0.909·5-s + 6·6-s − 16.3·7-s + 8·8-s + 9·9-s − 1.81·10-s − 1.94·11-s + 12·12-s + 15.0·13-s − 32.6·14-s − 2.72·15-s + 16·16-s + 32.0·17-s + 18·18-s − 3.63·20-s − 48.9·21-s − 3.88·22-s − 44.8·23-s + 24·24-s − 124.·25-s + 30.1·26-s + 27·27-s − 65.3·28-s + 31.1·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.0813·5-s + 0.408·6-s − 0.881·7-s + 0.353·8-s + 0.333·9-s − 0.0575·10-s − 0.0532·11-s + 0.288·12-s + 0.321·13-s − 0.623·14-s − 0.0469·15-s + 0.250·16-s + 0.456·17-s + 0.235·18-s − 0.0406·20-s − 0.508·21-s − 0.0376·22-s − 0.406·23-s + 0.204·24-s − 0.993·25-s + 0.227·26-s + 0.192·27-s − 0.440·28-s + 0.199·29-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: $\chi_{2166} (1, \cdot )$
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 \( 1 + 0.909T + 125T^{2} \)
7 \( 1 + 16.3T + 343T^{2} \)
11 \( 1 + 1.94T + 1.33e3T^{2} \)
13 \( 1 - 15.0T + 2.19e3T^{2} \)
17 \( 1 - 32.0T + 4.91e3T^{2} \)
23 \( 1 + 44.8T + 1.21e4T^{2} \)
29 \( 1 - 31.1T + 2.43e4T^{2} \)
31 \( 1 + 299.T + 2.97e4T^{2} \)
37 \( 1 + 252.T + 5.06e4T^{2} \)
41 \( 1 - 81.6T + 6.89e4T^{2} \)
43 \( 1 + 217.T + 7.95e4T^{2} \)
47 \( 1 - 187.T + 1.03e5T^{2} \)
53 \( 1 + 11.8T + 1.48e5T^{2} \)
59 \( 1 - 244.T + 2.05e5T^{2} \)
61 \( 1 - 526.T + 2.26e5T^{2} \)
67 \( 1 + 210.T + 3.00e5T^{2} \)
71 \( 1 + 503.T + 3.57e5T^{2} \)
73 \( 1 - 115.T + 3.89e5T^{2} \)
79 \( 1 - 220.T + 4.93e5T^{2} \)
83 \( 1 - 200.T + 5.71e5T^{2} \)
89 \( 1 + 174.T + 7.04e5T^{2} \)
97 \( 1 + 1.10e3T + 9.12e5T^{2} \)
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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

−8.263422755456844582697810102977, −7.45137071620747141844427692051, −6.75575822488754397896106522873, −5.90852378574887775809878217043, −5.17303782526881701013523669880, −3.92231868785952042072083790861, −3.54196768744657859336427383116, −2.55245501851747760710249486406, −1.54258784359168848706279111674, 0, 1.54258784359168848706279111674, 2.55245501851747760710249486406, 3.54196768744657859336427383116, 3.92231868785952042072083790861, 5.17303782526881701013523669880, 5.90852378574887775809878217043, 6.75575822488754397896106522873, 7.45137071620747141844427692051, 8.263422755456844582697810102977

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