Properties

Label 2-2166-1.1-c3-0-16
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 + 20.2·5-s + 6·6-s − 22.8·7-s − 8·8-s + 9·9-s − 40.4·10-s − 57.5·11-s − 12·12-s + 27.3·13-s + 45.7·14-s − 60.6·15-s + 16·16-s − 73.4·17-s − 18·18-s + 80.8·20-s + 68.6·21-s + 115.·22-s − 188.·23-s + 24·24-s + 283.·25-s − 54.6·26-s − 27·27-s − 91.5·28-s − 7.15·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.80·5-s + 0.408·6-s − 1.23·7-s − 0.353·8-s + 0.333·9-s − 1.27·10-s − 1.57·11-s − 0.288·12-s + 0.582·13-s + 0.873·14-s − 1.04·15-s + 0.250·16-s − 1.04·17-s − 0.235·18-s + 0.903·20-s + 0.713·21-s + 1.11·22-s − 1.70·23-s + 0.204·24-s + 2.26·25-s − 0.412·26-s − 0.192·27-s − 0.617·28-s − 0.0458·29-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

\(L(2)\) \(\approx\) \(0.8116690453\)
\(L(\frac12)\) \(\approx\) \(0.8116690453\)
\(L(\frac{5}{2})\) 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 \( 1 - 20.2T + 125T^{2} \)
7 \( 1 + 22.8T + 343T^{2} \)
11 \( 1 + 57.5T + 1.33e3T^{2} \)
13 \( 1 - 27.3T + 2.19e3T^{2} \)
17 \( 1 + 73.4T + 4.91e3T^{2} \)
23 \( 1 + 188.T + 1.21e4T^{2} \)
29 \( 1 + 7.15T + 2.43e4T^{2} \)
31 \( 1 + 117.T + 2.97e4T^{2} \)
37 \( 1 + 332.T + 5.06e4T^{2} \)
41 \( 1 + 42.7T + 6.89e4T^{2} \)
43 \( 1 - 47.2T + 7.95e4T^{2} \)
47 \( 1 + 507.T + 1.03e5T^{2} \)
53 \( 1 - 492.T + 1.48e5T^{2} \)
59 \( 1 - 460.T + 2.05e5T^{2} \)
61 \( 1 - 450.T + 2.26e5T^{2} \)
67 \( 1 - 522.T + 3.00e5T^{2} \)
71 \( 1 - 931.T + 3.57e5T^{2} \)
73 \( 1 - 350.T + 3.89e5T^{2} \)
79 \( 1 - 194.T + 4.93e5T^{2} \)
83 \( 1 + 286.T + 5.71e5T^{2} \)
89 \( 1 - 839.T + 7.04e5T^{2} \)
97 \( 1 - 605.T + 9.12e5T^{2} \)
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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

−8.903121366967599808418428692172, −8.080913927152011127705207480951, −6.83880998781424743445759696516, −6.46609059890285077924227828719, −5.67803190946944679896940016732, −5.17021378162092595996187648728, −3.60796648826634215972802042430, −2.42869276302355518498675072260, −1.89622152809635654764390417230, −0.44095997027413699451940590790, 0.44095997027413699451940590790, 1.89622152809635654764390417230, 2.42869276302355518498675072260, 3.60796648826634215972802042430, 5.17021378162092595996187648728, 5.67803190946944679896940016732, 6.46609059890285077924227828719, 6.83880998781424743445759696516, 8.080913927152011127705207480951, 8.903121366967599808418428692172

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