Properties

Label 2-2166-1.1-c1-0-39
Degree $2$
Conductor $2166$
Sign $-1$
Analytic cond. $17.2955$
Root an. cond. $4.15879$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 1.18·5-s + 6-s + 0.532·7-s − 8-s + 9-s − 1.18·10-s + 1.87·11-s − 12-s − 3.87·13-s − 0.532·14-s − 1.18·15-s + 16-s + 1.16·17-s − 18-s + 1.18·20-s − 0.532·21-s − 1.87·22-s − 6.70·23-s + 24-s − 3.59·25-s + 3.87·26-s − 27-s + 0.532·28-s − 4.02·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.529·5-s + 0.408·6-s + 0.201·7-s − 0.353·8-s + 0.333·9-s − 0.374·10-s + 0.566·11-s − 0.288·12-s − 1.07·13-s − 0.142·14-s − 0.305·15-s + 0.250·16-s + 0.281·17-s − 0.235·18-s + 0.264·20-s − 0.116·21-s − 0.400·22-s − 1.39·23-s + 0.204·24-s − 0.719·25-s + 0.760·26-s − 0.192·27-s + 0.100·28-s − 0.746·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2166\)    =    \(2 \cdot 3 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(17.2955\)
Root analytic conductor: \(4.15879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2166,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
19 \( 1 \)
good5 \( 1 - 1.18T + 5T^{2} \)
7 \( 1 - 0.532T + 7T^{2} \)
11 \( 1 - 1.87T + 11T^{2} \)
13 \( 1 + 3.87T + 13T^{2} \)
17 \( 1 - 1.16T + 17T^{2} \)
23 \( 1 + 6.70T + 23T^{2} \)
29 \( 1 + 4.02T + 29T^{2} \)
31 \( 1 + 1.95T + 31T^{2} \)
37 \( 1 - 6.88T + 37T^{2} \)
41 \( 1 + 8.98T + 41T^{2} \)
43 \( 1 - 2.42T + 43T^{2} \)
47 \( 1 - 2.04T + 47T^{2} \)
53 \( 1 - 12.9T + 53T^{2} \)
59 \( 1 + 2.68T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 - 6.07T + 71T^{2} \)
73 \( 1 + 0.327T + 73T^{2} \)
79 \( 1 + 16.1T + 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 + 3.55T + 89T^{2} \)
97 \( 1 - 5.87T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.811624079176334604802416949093, −7.78918456147384585372173394774, −7.28704779297971240863843621219, −6.26289952371780842150357259451, −5.75177842599285797101495201610, −4.76738946898824609560017633027, −3.74148417606935645520749601608, −2.36786985556159669651503031299, −1.51343651624321780601306815271, 0, 1.51343651624321780601306815271, 2.36786985556159669651503031299, 3.74148417606935645520749601608, 4.76738946898824609560017633027, 5.75177842599285797101495201610, 6.26289952371780842150357259451, 7.28704779297971240863843621219, 7.78918456147384585372173394774, 8.811624079176334604802416949093

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