Properties

Label 2-33-1.1-c3-0-2
Degree $2$
Conductor $33$
Sign $-1$
Analytic cond. $1.94706$
Root an. cond. $1.39537$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s + 3·3-s + 17·4-s − 14·5-s − 15·6-s − 32·7-s − 45·8-s + 9·9-s + 70·10-s − 11·11-s + 51·12-s − 38·13-s + 160·14-s − 42·15-s + 89·16-s − 2·17-s − 45·18-s + 72·19-s − 238·20-s − 96·21-s + 55·22-s + 68·23-s − 135·24-s + 71·25-s + 190·26-s + 27·27-s − 544·28-s + ⋯
L(s)  = 1  − 1.76·2-s + 0.577·3-s + 17/8·4-s − 1.25·5-s − 1.02·6-s − 1.72·7-s − 1.98·8-s + 1/3·9-s + 2.21·10-s − 0.301·11-s + 1.22·12-s − 0.810·13-s + 3.05·14-s − 0.722·15-s + 1.39·16-s − 0.0285·17-s − 0.589·18-s + 0.869·19-s − 2.66·20-s − 0.997·21-s + 0.533·22-s + 0.616·23-s − 1.14·24-s + 0.567·25-s + 1.43·26-s + 0.192·27-s − 3.67·28-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 33,\ (\ :3/2),\ -1)\)

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)$
bad3 \( 1 - p T \)
11 \( 1 + p T \)
good2 \( 1 + 5 T + p^{3} T^{2} \)
5 \( 1 + 14 T + p^{3} T^{2} \)
7 \( 1 + 32 T + p^{3} T^{2} \)
13 \( 1 + 38 T + p^{3} T^{2} \)
17 \( 1 + 2 T + p^{3} T^{2} \)
19 \( 1 - 72 T + p^{3} T^{2} \)
23 \( 1 - 68 T + p^{3} T^{2} \)
29 \( 1 + 54 T + p^{3} T^{2} \)
31 \( 1 + 152 T + p^{3} T^{2} \)
37 \( 1 - 174 T + p^{3} T^{2} \)
41 \( 1 - 94 T + p^{3} T^{2} \)
43 \( 1 + 528 T + p^{3} T^{2} \)
47 \( 1 + 340 T + p^{3} T^{2} \)
53 \( 1 + 438 T + p^{3} T^{2} \)
59 \( 1 - 20 T + p^{3} T^{2} \)
61 \( 1 - 570 T + p^{3} T^{2} \)
67 \( 1 + 460 T + p^{3} T^{2} \)
71 \( 1 + 1092 T + p^{3} T^{2} \)
73 \( 1 - 562 T + p^{3} T^{2} \)
79 \( 1 + 16 T + p^{3} T^{2} \)
83 \( 1 - 372 T + p^{3} T^{2} \)
89 \( 1 + 966 T + p^{3} T^{2} \)
97 \( 1 + 526 T + p^{3} T^{2} \)
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

−16.01373331971803535604520658707, −15.10326458051728747934703527497, −12.89153469052912144347423721556, −11.56715041911891389839645523290, −10.05912369008524940219976852127, −9.204124258956680265368272925335, −7.83502351517021965034170796576, −6.91749548994513994727963273880, −3.14217182018133821800082250424, 0, 3.14217182018133821800082250424, 6.91749548994513994727963273880, 7.83502351517021965034170796576, 9.204124258956680265368272925335, 10.05912369008524940219976852127, 11.56715041911891389839645523290, 12.89153469052912144347423721556, 15.10326458051728747934703527497, 16.01373331971803535604520658707

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