Properties

Label 2-33-1.1-c5-0-7
Degree $2$
Conductor $33$
Sign $-1$
Analytic cond. $5.29266$
Root an. cond. $2.30057$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 4.15·2-s − 9·3-s − 14.7·4-s − 37.5·5-s − 37.3·6-s − 76.4·7-s − 194.·8-s + 81·9-s − 155.·10-s − 121·11-s + 132.·12-s + 169.·13-s − 317.·14-s + 337.·15-s − 333.·16-s − 0.875·17-s + 336.·18-s − 817.·19-s + 553.·20-s + 688.·21-s − 502.·22-s + 749.·23-s + 1.74e3·24-s − 1.71e3·25-s + 704.·26-s − 729·27-s + 1.12e3·28-s + ⋯
L(s)  = 1  + 0.733·2-s − 0.577·3-s − 0.461·4-s − 0.671·5-s − 0.423·6-s − 0.589·7-s − 1.07·8-s + 0.333·9-s − 0.492·10-s − 0.301·11-s + 0.266·12-s + 0.278·13-s − 0.433·14-s + 0.387·15-s − 0.325·16-s − 0.000735·17-s + 0.244·18-s − 0.519·19-s + 0.309·20-s + 0.340·21-s − 0.221·22-s + 0.295·23-s + 0.619·24-s − 0.549·25-s + 0.204·26-s − 0.192·27-s + 0.272·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-1$
Analytic conductor: \(5.29266\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9T \)
11 \( 1 + 121T \)
good2 \( 1 - 4.15T + 32T^{2} \)
5 \( 1 + 37.5T + 3.12e3T^{2} \)
7 \( 1 + 76.4T + 1.68e4T^{2} \)
13 \( 1 - 169.T + 3.71e5T^{2} \)
17 \( 1 + 0.875T + 1.41e6T^{2} \)
19 \( 1 + 817.T + 2.47e6T^{2} \)
23 \( 1 - 749.T + 6.43e6T^{2} \)
29 \( 1 - 6.04e3T + 2.05e7T^{2} \)
31 \( 1 + 1.47e3T + 2.86e7T^{2} \)
37 \( 1 + 1.58e4T + 6.93e7T^{2} \)
41 \( 1 + 7.62e3T + 1.15e8T^{2} \)
43 \( 1 + 1.82e4T + 1.47e8T^{2} \)
47 \( 1 + 1.28e4T + 2.29e8T^{2} \)
53 \( 1 - 2.17e4T + 4.18e8T^{2} \)
59 \( 1 + 1.16e3T + 7.14e8T^{2} \)
61 \( 1 - 1.40e4T + 8.44e8T^{2} \)
67 \( 1 - 3.69e4T + 1.35e9T^{2} \)
71 \( 1 + 3.75e4T + 1.80e9T^{2} \)
73 \( 1 - 8.04e4T + 2.07e9T^{2} \)
79 \( 1 + 6.21e4T + 3.07e9T^{2} \)
83 \( 1 + 1.19e4T + 3.93e9T^{2} \)
89 \( 1 - 1.46e5T + 5.58e9T^{2} \)
97 \( 1 + 1.38e5T + 8.58e9T^{2} \)
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

−15.13360148874937302477256187810, −13.69412194241532301953161295221, −12.66574831874663081872849108495, −11.68869704591268006977706955129, −10.11923600090473649603053583263, −8.491670798461644446160255407481, −6.58098047892767702440617356094, −5.05448953737723395622846367728, −3.58137809057447407668855622201, 0, 3.58137809057447407668855622201, 5.05448953737723395622846367728, 6.58098047892767702440617356094, 8.491670798461644446160255407481, 10.11923600090473649603053583263, 11.68869704591268006977706955129, 12.66574831874663081872849108495, 13.69412194241532301953161295221, 15.13360148874937302477256187810

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