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Properties

Label 2-33-11.10-c2-0-3
Degree $2$
Conductor $33$
Sign $-0.612 + 0.790i$
Analytic cond. $0.899184$
Root an. cond. $0.948253$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.53i·2-s − 1.73·3-s − 8.46·4-s + 6.19·5-s + 6.11i·6-s − 2.58i·7-s + 15.7i·8-s + 2.99·9-s − 21.8i·10-s + (6.73 − 8.69i)11-s + 14.6·12-s + 23.7i·13-s − 9.12·14-s − 10.7·15-s + 21.7·16-s + 12.2i·17-s + ⋯
L(s)  = 1  − 1.76i·2-s − 0.577·3-s − 2.11·4-s + 1.23·5-s + 1.01i·6-s − 0.369i·7-s + 1.97i·8-s + 0.333·9-s − 2.18i·10-s + (0.612 − 0.790i)11-s + 1.22·12-s + 1.82i·13-s − 0.651·14-s − 0.715·15-s + 1.36·16-s + 0.719i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.612 + 0.790i$
Analytic conductor: \(0.899184\)
Root analytic conductor: \(0.948253\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1),\ -0.612 + 0.790i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.404509 - 0.824515i\)
\(L(\frac12)\) \(\approx\) \(0.404509 - 0.824515i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
11 \( 1 + (-6.73 + 8.69i)T \)
good2 \( 1 + 3.53iT - 4T^{2} \)
5 \( 1 - 6.19T + 25T^{2} \)
7 \( 1 + 2.58iT - 49T^{2} \)
13 \( 1 - 23.7iT - 169T^{2} \)
17 \( 1 - 12.2iT - 289T^{2} \)
19 \( 1 - 3.27iT - 361T^{2} \)
23 \( 1 + 14.3T + 529T^{2} \)
29 \( 1 + 38.5iT - 841T^{2} \)
31 \( 1 + 11.1T + 961T^{2} \)
37 \( 1 + 12.5T + 1.36e3T^{2} \)
41 \( 1 + 1.38iT - 1.68e3T^{2} \)
43 \( 1 - 23.9iT - 1.84e3T^{2} \)
47 \( 1 - 19.8T + 2.20e3T^{2} \)
53 \( 1 + 12.0T + 2.80e3T^{2} \)
59 \( 1 + 62.7T + 3.48e3T^{2} \)
61 \( 1 + 21.3iT - 3.72e3T^{2} \)
67 \( 1 + 34T + 4.48e3T^{2} \)
71 \( 1 + 69.2T + 5.04e3T^{2} \)
73 \( 1 + 39.9iT - 5.32e3T^{2} \)
79 \( 1 - 97.6iT - 6.24e3T^{2} \)
83 \( 1 + 71.9iT - 6.88e3T^{2} \)
89 \( 1 - 107.T + 7.92e3T^{2} \)
97 \( 1 + 166.T + 9.40e3T^{2} \)
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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

−16.66089905828759549416081151807, −14.09977699405630018722588118712, −13.49955232345092999990011623682, −12.10798662709275078093249925492, −11.14299715677443179220938044031, −10.01851215469821635860244540049, −9.069322791113611605464269711606, −6.15012025980602209124338683622, −4.15979940465240215847716423614, −1.75980946077415687813493024952, 5.14629586213630255833811611530, 6.01785272799938372427563424404, 7.34390685369888332716055361569, 9.027180936759949181677452379217, 10.20382368798121601978165258507, 12.52807099055674813784751832003, 13.66903638690475769431315642723, 14.80850459626339352154425597640, 15.78602266977772422524583427412, 16.98468076297493011220349814447

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