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Properties

Label 2-33-11.3-c1-0-0
Degree $2$
Conductor $33$
Sign $0.394 - 0.918i$
Analytic cond. $0.263506$
Root an. cond. $0.513328$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 + 0.951i)2-s + (0.309 + 0.951i)3-s + (0.190 − 0.587i)4-s + (0.309 + 0.224i)5-s + (−1.30 − 0.951i)6-s + (0.927 − 2.85i)7-s + (−0.690 − 2.12i)8-s + (−0.809 + 0.587i)9-s − 0.618·10-s + (2.80 − 1.76i)11-s + 0.618·12-s + (−5.04 + 3.66i)13-s + (1.5 + 4.61i)14-s + (−0.118 + 0.363i)15-s + (3.92 + 2.85i)16-s + (0.5 + 0.363i)17-s + ⋯
L(s)  = 1  + (−0.925 + 0.672i)2-s + (0.178 + 0.549i)3-s + (0.0954 − 0.293i)4-s + (0.138 + 0.100i)5-s + (−0.534 − 0.388i)6-s + (0.350 − 1.07i)7-s + (−0.244 − 0.751i)8-s + (−0.269 + 0.195i)9-s − 0.195·10-s + (0.846 − 0.531i)11-s + 0.178·12-s + (−1.39 + 1.01i)13-s + (0.400 + 1.23i)14-s + (−0.0304 + 0.0937i)15-s + (0.981 + 0.713i)16-s + (0.121 + 0.0881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.394 - 0.918i$
Analytic conductor: \(0.263506\)
Root analytic conductor: \(0.513328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1/2),\ 0.394 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.428985 + 0.282629i\)
\(L(\frac12)\) \(\approx\) \(0.428985 + 0.282629i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-2.80 + 1.76i)T \)
good2 \( 1 + (1.30 - 0.951i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-0.309 - 0.224i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.927 + 2.85i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (5.04 - 3.66i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.5 - 0.363i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.263 + 0.812i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 5.47T + 23T^{2} \)
29 \( 1 + (1.38 - 4.25i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.11 + 2.26i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.30 - 4.02i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.83 - 5.65i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.76T + 43T^{2} \)
47 \( 1 + (0.190 + 0.587i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.97 + 4.33i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.64 - 5.06i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.927 + 0.673i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 + (11.7 + 8.55i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.381 + 1.17i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.427 - 0.310i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-10.2 - 7.46i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 9.47T + 89T^{2} \)
97 \( 1 + (12.1 - 8.83i)T + (29.9 - 92.2i)T^{2} \)
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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

−16.87329074195413596442382606279, −16.31913237700596334324519217137, −14.76110885406189097238077134206, −13.83885941459867377969161228911, −11.91238724455998699638727578856, −10.27991495870979554002009742288, −9.311555208116823992380309691437, −7.935894424972931507535866257347, −6.65892509575430838722444846410, −4.17616876775155368877715514137, 2.21987324717829730221720149871, 5.57004126017888279944990882501, 7.73089604144674073321104060504, 9.020062748017145480790623808267, 10.04425102563079848458115774396, 11.72925486511160377615403281474, 12.42794079269473768134228259268, 14.29817984089071462665826097451, 15.22487100736871192818418686697, 17.27249853756645298396465979323

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