Properties

Label 2-2166-1.1-c3-0-148
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 − 12.0·5-s + 6·6-s − 0.673·7-s + 8·8-s + 9·9-s − 24.1·10-s + 15.9·11-s + 12·12-s + 3.90·13-s − 1.34·14-s − 36.2·15-s + 16·16-s − 44.0·17-s + 18·18-s − 48.3·20-s − 2.02·21-s + 31.8·22-s − 9.11·23-s + 24·24-s + 21.1·25-s + 7.81·26-s + 27·27-s − 2.69·28-s − 161.·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.08·5-s + 0.408·6-s − 0.0363·7-s + 0.353·8-s + 0.333·9-s − 0.764·10-s + 0.437·11-s + 0.288·12-s + 0.0834·13-s − 0.0257·14-s − 0.624·15-s + 0.250·16-s − 0.627·17-s + 0.235·18-s − 0.540·20-s − 0.0210·21-s + 0.309·22-s − 0.0826·23-s + 0.204·24-s + 0.169·25-s + 0.0589·26-s + 0.192·27-s − 0.0181·28-s − 1.03·29-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 \( 1 + 12.0T + 125T^{2} \)
7 \( 1 + 0.673T + 343T^{2} \)
11 \( 1 - 15.9T + 1.33e3T^{2} \)
13 \( 1 - 3.90T + 2.19e3T^{2} \)
17 \( 1 + 44.0T + 4.91e3T^{2} \)
23 \( 1 + 9.11T + 1.21e4T^{2} \)
29 \( 1 + 161.T + 2.43e4T^{2} \)
31 \( 1 - 60.5T + 2.97e4T^{2} \)
37 \( 1 - 104.T + 5.06e4T^{2} \)
41 \( 1 - 249.T + 6.89e4T^{2} \)
43 \( 1 + 6.90T + 7.95e4T^{2} \)
47 \( 1 - 145.T + 1.03e5T^{2} \)
53 \( 1 + 754.T + 1.48e5T^{2} \)
59 \( 1 + 704.T + 2.05e5T^{2} \)
61 \( 1 + 32.5T + 2.26e5T^{2} \)
67 \( 1 - 77.7T + 3.00e5T^{2} \)
71 \( 1 - 44.4T + 3.57e5T^{2} \)
73 \( 1 - 280.T + 3.89e5T^{2} \)
79 \( 1 - 779.T + 4.93e5T^{2} \)
83 \( 1 + 274.T + 5.71e5T^{2} \)
89 \( 1 + 1.00e3T + 7.04e5T^{2} \)
97 \( 1 + 832.T + 9.12e5T^{2} \)
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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

−8.073453946281869208373139998544, −7.66205325662059052464281394390, −6.77492338296616773864077399924, −6.02506408762606047336468711322, −4.84176253361136468980567202819, −4.13694037839207776881827290322, −3.52262976430095927985305369391, −2.59916512531750744566313391627, −1.45363021171602645785431643037, 0, 1.45363021171602645785431643037, 2.59916512531750744566313391627, 3.52262976430095927985305369391, 4.13694037839207776881827290322, 4.84176253361136468980567202819, 6.02506408762606047336468711322, 6.77492338296616773864077399924, 7.66205325662059052464281394390, 8.073453946281869208373139998544

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