Properties

Label 2-33-1.1-c3-0-4
Degree $2$
Conductor $33$
Sign $1$
Analytic cond. $1.94706$
Root an. cond. $1.39537$
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  + 5.42·2-s − 3·3-s + 21.4·4-s − 16.8·5-s − 16.2·6-s − 7.69·7-s + 72.8·8-s + 9·9-s − 91.3·10-s − 11·11-s − 64.2·12-s + 24.8·13-s − 41.7·14-s + 50.5·15-s + 223.·16-s − 15.9·17-s + 48.8·18-s + 15.1·19-s − 360.·20-s + 23.0·21-s − 59.6·22-s + 17.7·23-s − 218.·24-s + 158.·25-s + 134.·26-s − 27·27-s − 164.·28-s + ⋯
L(s)  = 1  + 1.91·2-s − 0.577·3-s + 2.67·4-s − 1.50·5-s − 1.10·6-s − 0.415·7-s + 3.21·8-s + 0.333·9-s − 2.89·10-s − 0.301·11-s − 1.54·12-s + 0.530·13-s − 0.797·14-s + 0.870·15-s + 3.49·16-s − 0.227·17-s + 0.639·18-s + 0.182·19-s − 4.03·20-s + 0.239·21-s − 0.578·22-s + 0.160·23-s − 1.85·24-s + 1.27·25-s + 1.01·26-s − 0.192·27-s − 1.11·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $1$
Analytic conductor: \(1.94706\)
Root analytic conductor: \(1.39537\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.402021435\)
\(L(\frac12)\) \(\approx\) \(2.402021435\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
11 \( 1 + 11T \)
good2 \( 1 - 5.42T + 8T^{2} \)
5 \( 1 + 16.8T + 125T^{2} \)
7 \( 1 + 7.69T + 343T^{2} \)
13 \( 1 - 24.8T + 2.19e3T^{2} \)
17 \( 1 + 15.9T + 4.91e3T^{2} \)
19 \( 1 - 15.1T + 6.85e3T^{2} \)
23 \( 1 - 17.7T + 1.21e4T^{2} \)
29 \( 1 + 128.T + 2.43e4T^{2} \)
31 \( 1 - 219.T + 2.97e4T^{2} \)
37 \( 1 - 92.0T + 5.06e4T^{2} \)
41 \( 1 + 459.T + 6.89e4T^{2} \)
43 \( 1 - 64.9T + 7.95e4T^{2} \)
47 \( 1 - 497.T + 1.03e5T^{2} \)
53 \( 1 + 526.T + 1.48e5T^{2} \)
59 \( 1 + 578.T + 2.05e5T^{2} \)
61 \( 1 + 221.T + 2.26e5T^{2} \)
67 \( 1 + 860.T + 3.00e5T^{2} \)
71 \( 1 - 580.T + 3.57e5T^{2} \)
73 \( 1 - 510.T + 3.89e5T^{2} \)
79 \( 1 - 1.03e3T + 4.93e5T^{2} \)
83 \( 1 - 606.T + 5.71e5T^{2} \)
89 \( 1 + 23.4T + 7.04e5T^{2} \)
97 \( 1 - 719.T + 9.12e5T^{2} \)
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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

−15.71459721248052431777341977517, −15.22139744115513506306848104642, −13.65385847386436529128082517717, −12.52751069401484390402409559939, −11.69814408747768663976064065374, −10.79310108261216811024923539595, −7.66190969758609841466684102712, −6.35523633698523852422200744919, −4.73356142042452570333742314381, −3.45814931002554557885865185437, 3.45814931002554557885865185437, 4.73356142042452570333742314381, 6.35523633698523852422200744919, 7.66190969758609841466684102712, 10.79310108261216811024923539595, 11.69814408747768663976064065374, 12.52751069401484390402409559939, 13.65385847386436529128082517717, 15.22139744115513506306848104642, 15.71459721248052431777341977517

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