Properties

Label 2-33-1.1-c3-0-5
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s − 7·4-s − 4·5-s + 3·6-s − 26·7-s + 15·8-s + 9·9-s + 4·10-s + 11·11-s + 21·12-s − 32·13-s + 26·14-s + 12·15-s + 41·16-s + 74·17-s − 9·18-s − 60·19-s + 28·20-s + 78·21-s − 11·22-s − 182·23-s − 45·24-s − 109·25-s + 32·26-s − 27·27-s + 182·28-s + ⋯
L(s)  = 1  − 0.353·2-s − 0.577·3-s − 7/8·4-s − 0.357·5-s + 0.204·6-s − 1.40·7-s + 0.662·8-s + 1/3·9-s + 0.126·10-s + 0.301·11-s + 0.505·12-s − 0.682·13-s + 0.496·14-s + 0.206·15-s + 0.640·16-s + 1.05·17-s − 0.117·18-s − 0.724·19-s + 0.313·20-s + 0.810·21-s − 0.106·22-s − 1.64·23-s − 0.382·24-s − 0.871·25-s + 0.241·26-s − 0.192·27-s + 1.22·28-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 \( 1 + T + p^{3} T^{2} \)
5 \( 1 + 4 T + p^{3} T^{2} \)
7 \( 1 + 26 T + p^{3} T^{2} \)
13 \( 1 + 32 T + p^{3} T^{2} \)
17 \( 1 - 74 T + p^{3} T^{2} \)
19 \( 1 + 60 T + p^{3} T^{2} \)
23 \( 1 + 182 T + p^{3} T^{2} \)
29 \( 1 + 90 T + p^{3} T^{2} \)
31 \( 1 + 8 T + p^{3} T^{2} \)
37 \( 1 + 66 T + p^{3} T^{2} \)
41 \( 1 - 422 T + p^{3} T^{2} \)
43 \( 1 - 408 T + p^{3} T^{2} \)
47 \( 1 + 506 T + p^{3} T^{2} \)
53 \( 1 - 348 T + p^{3} T^{2} \)
59 \( 1 + 200 T + p^{3} T^{2} \)
61 \( 1 - 132 T + p^{3} T^{2} \)
67 \( 1 + 1036 T + p^{3} T^{2} \)
71 \( 1 - 762 T + p^{3} T^{2} \)
73 \( 1 + 542 T + p^{3} T^{2} \)
79 \( 1 + 550 T + p^{3} T^{2} \)
83 \( 1 + 132 T + p^{3} T^{2} \)
89 \( 1 - 570 T + p^{3} T^{2} \)
97 \( 1 - 14 T + p^{3} T^{2} \)
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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

−15.99800527471375409681628617840, −14.38916713662311871880750714929, −13.00270062249319427861056346089, −12.08948336874890052910498086982, −10.24583505954726159699044737823, −9.411879972450193940857107479417, −7.67922620982539398739813089313, −5.93154960330735044974422265346, −3.99071285874632850409323170230, 0, 3.99071285874632850409323170230, 5.93154960330735044974422265346, 7.67922620982539398739813089313, 9.411879972450193940857107479417, 10.24583505954726159699044737823, 12.08948336874890052910498086982, 13.00270062249319427861056346089, 14.38916713662311871880750714929, 15.99800527471375409681628617840

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