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Properties

Label 2-33-33.29-c1-0-0
Degree $2$
Conductor $33$
Sign $0.942 + 0.334i$
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  + (−0.951 − 0.690i)2-s + (1.34 + 1.08i)3-s + (−0.190 − 0.587i)4-s + (−0.224 − 0.309i)5-s + (−0.530 − 1.96i)6-s + (−2.92 + 0.951i)7-s + (−0.951 + 2.92i)8-s + (0.633 + 2.93i)9-s + 0.449i·10-s + (−1.31 − 3.04i)11-s + (0.381 − 1.00i)12-s + (0.427 − 0.587i)13-s + (3.44 + 1.11i)14-s + (0.0335 − 0.660i)15-s + (1.92 − 1.40i)16-s + (3.44 − 2.5i)17-s + ⋯
L(s)  = 1  + (−0.672 − 0.488i)2-s + (0.778 + 0.628i)3-s + (−0.0954 − 0.293i)4-s + (−0.100 − 0.138i)5-s + (−0.216 − 0.802i)6-s + (−1.10 + 0.359i)7-s + (−0.336 + 1.03i)8-s + (0.211 + 0.977i)9-s + 0.141i·10-s + (−0.396 − 0.918i)11-s + (0.110 − 0.288i)12-s + (0.118 − 0.163i)13-s + (0.919 + 0.298i)14-s + (0.00865 − 0.170i)15-s + (0.481 − 0.350i)16-s + (0.834 − 0.606i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.942 + 0.334i$
Analytic conductor: \(0.263506\)
Root analytic conductor: \(0.513328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1/2),\ 0.942 + 0.334i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.586808 - 0.101058i\)
\(L(\frac12)\) \(\approx\) \(0.586808 - 0.101058i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.34 - 1.08i)T \)
11 \( 1 + (1.31 + 3.04i)T \)
good2 \( 1 + (0.951 + 0.690i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (0.224 + 0.309i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (2.92 - 0.951i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.427 + 0.587i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.44 + 2.5i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.5 - 0.812i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 6.23iT - 23T^{2} \)
29 \( 1 + (0.726 + 2.23i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.73 + 3.44i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.927 - 2.85i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.224 + 0.690i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 6.88iT - 43T^{2} \)
47 \( 1 + (7.91 + 2.57i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.08 + 1.5i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.363 - 0.118i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.54 - 2.12i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 7.32T + 67T^{2} \)
71 \( 1 + (-3.13 - 4.30i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (8.94 - 2.90i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.33 + 3.21i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.11 - 3.71i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 0.527iT - 89T^{2} \)
97 \( 1 + (-3.54 - 2.57i)T + (29.9 + 92.2i)T^{2} \)
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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

−16.52407368860401578799669827456, −15.69162057670535242444503001389, −14.37028360535132920080517534857, −13.24406638195257779152916154982, −11.44511872089849842837755744737, −10.04152078806631209916398546952, −9.349267910346159085418587859481, −8.079194580715701290648316969186, −5.56387926442212758414465247152, −3.09946550432729284409509744350, 3.44425920555600456748392683993, 6.72249174347247341526734036687, 7.60357455228223250295743776746, 8.984889786304366107862782126255, 10.05877927778640596705161843897, 12.46111477857325269655280671237, 13.05209789288985809445159770658, 14.59060497885050493016777362217, 15.83485358740221133375987018016, 16.89256020590694664833991793915

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