Properties

Degree $2$
Conductor $33$
Sign $-1$
Motivic weight $5$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 9·3-s − 31·4-s − 92·5-s + 9·6-s − 26·7-s − 63·8-s + 81·9-s − 92·10-s + 121·11-s − 279·12-s − 692·13-s − 26·14-s − 828·15-s + 929·16-s − 1.44e3·17-s + 81·18-s + 2.16e3·19-s + 2.85e3·20-s − 234·21-s + 121·22-s − 1.58e3·23-s − 567·24-s + 5.33e3·25-s − 692·26-s + 729·27-s + 806·28-s + ⋯
L(s)  = 1  + 0.176·2-s + 0.577·3-s − 0.968·4-s − 1.64·5-s + 0.102·6-s − 0.200·7-s − 0.348·8-s + 1/3·9-s − 0.290·10-s + 0.301·11-s − 0.559·12-s − 1.13·13-s − 0.0354·14-s − 0.950·15-s + 0.907·16-s − 1.21·17-s + 0.0589·18-s + 1.37·19-s + 1.59·20-s − 0.115·21-s + 0.0533·22-s − 0.623·23-s − 0.200·24-s + 1.70·25-s − 0.200·26-s + 0.192·27-s + 0.194·28-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: $-1$
Motivic weight: \(5\)
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 \( 1 - p^{2} T \)
11 \( 1 - p^{2} T \)
good2 \( 1 - T + p^{5} T^{2} \)
5 \( 1 + 92 T + p^{5} T^{2} \)
7 \( 1 + 26 T + p^{5} T^{2} \)
13 \( 1 + 692 T + p^{5} T^{2} \)
17 \( 1 + 1442 T + p^{5} T^{2} \)
19 \( 1 - 2160 T + p^{5} T^{2} \)
23 \( 1 + 1582 T + p^{5} T^{2} \)
29 \( 1 + 5526 T + p^{5} T^{2} \)
31 \( 1 - 4792 T + p^{5} T^{2} \)
37 \( 1 + 10194 T + p^{5} T^{2} \)
41 \( 1 + 10622 T + p^{5} T^{2} \)
43 \( 1 - 8580 T + p^{5} T^{2} \)
47 \( 1 + 2362 T + p^{5} T^{2} \)
53 \( 1 + 30804 T + p^{5} T^{2} \)
59 \( 1 - 6416 T + p^{5} T^{2} \)
61 \( 1 - 42096 T + p^{5} T^{2} \)
67 \( 1 + 28444 T + p^{5} T^{2} \)
71 \( 1 - 45690 T + p^{5} T^{2} \)
73 \( 1 + 18374 T + p^{5} T^{2} \)
79 \( 1 + 105214 T + p^{5} T^{2} \)
83 \( 1 - 62292 T + p^{5} T^{2} \)
89 \( 1 + 72246 T + p^{5} T^{2} \)
97 \( 1 - 79262 T + p^{5} T^{2} \)
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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

−15.04545088187470708793847680822, −13.97906495770982178781141224417, −12.65131546278528974072539142568, −11.60688268578093135515933232181, −9.701909360678696774231479945700, −8.461571336702697651705839541929, −7.31976147177241080382725337192, −4.69833549041135197409722948204, −3.48454727351222856936721607244, 0, 3.48454727351222856936721607244, 4.69833549041135197409722948204, 7.31976147177241080382725337192, 8.461571336702697651705839541929, 9.701909360678696774231479945700, 11.60688268578093135515933232181, 12.65131546278528974072539142568, 13.97906495770982178781141224417, 15.04545088187470708793847680822

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