Properties

Label 2-2166-1.1-c3-0-42
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 4·4-s − 5.21·5-s + 6·6-s + 31.4·7-s − 8·8-s + 9·9-s + 10.4·10-s − 21.2·11-s − 12·12-s + 56.2·13-s − 62.9·14-s + 15.6·15-s + 16·16-s + 17.2·17-s − 18·18-s − 20.8·20-s − 94.4·21-s + 42.4·22-s − 206.·23-s + 24·24-s − 97.8·25-s − 112.·26-s − 27·27-s + 125.·28-s + 206.·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.466·5-s + 0.408·6-s + 1.70·7-s − 0.353·8-s + 0.333·9-s + 0.329·10-s − 0.581·11-s − 0.288·12-s + 1.20·13-s − 1.20·14-s + 0.269·15-s + 0.250·16-s + 0.246·17-s − 0.235·18-s − 0.233·20-s − 0.981·21-s + 0.411·22-s − 1.87·23-s + 0.204·24-s − 0.782·25-s − 0.848·26-s − 0.192·27-s + 0.850·28-s + 1.32·29-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: \(0\)
Selberg data: \((2,\ 2166,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.498456331\)
\(L(\frac12)\) \(\approx\) \(1.498456331\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.21T + 125T^{2} \)
7 \( 1 - 31.4T + 343T^{2} \)
11 \( 1 + 21.2T + 1.33e3T^{2} \)
13 \( 1 - 56.2T + 2.19e3T^{2} \)
17 \( 1 - 17.2T + 4.91e3T^{2} \)
23 \( 1 + 206.T + 1.21e4T^{2} \)
29 \( 1 - 206.T + 2.43e4T^{2} \)
31 \( 1 - 129.T + 2.97e4T^{2} \)
37 \( 1 - 440.T + 5.06e4T^{2} \)
41 \( 1 - 435.T + 6.89e4T^{2} \)
43 \( 1 - 129.T + 7.95e4T^{2} \)
47 \( 1 - 108.T + 1.03e5T^{2} \)
53 \( 1 + 407.T + 1.48e5T^{2} \)
59 \( 1 + 116.T + 2.05e5T^{2} \)
61 \( 1 + 340.T + 2.26e5T^{2} \)
67 \( 1 + 210.T + 3.00e5T^{2} \)
71 \( 1 - 158.T + 3.57e5T^{2} \)
73 \( 1 - 573.T + 3.89e5T^{2} \)
79 \( 1 + 885.T + 4.93e5T^{2} \)
83 \( 1 + 573.T + 5.71e5T^{2} \)
89 \( 1 - 215.T + 7.04e5T^{2} \)
97 \( 1 - 528.T + 9.12e5T^{2} \)
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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

−8.417089315952274553481846403880, −7.945591642889218733826487826952, −7.60537801940669381003094039123, −6.22773854425556557886805027433, −5.78825269029713768770666533443, −4.61434593765752845778438261117, −4.05113348020977982625411338437, −2.55954501782675289260351456091, −1.51416382320024480348750390503, −0.68727987613307161852698614907, 0.68727987613307161852698614907, 1.51416382320024480348750390503, 2.55954501782675289260351456091, 4.05113348020977982625411338437, 4.61434593765752845778438261117, 5.78825269029713768770666533443, 6.22773854425556557886805027433, 7.60537801940669381003094039123, 7.945591642889218733826487826952, 8.417089315952274553481846403880

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