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Properties

Label 2-33-11.3-c1-0-1
Degree $2$
Conductor $33$
Sign $0.970 + 0.242i$
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  + (0.309 − 0.224i)2-s + (−0.309 − 0.951i)3-s + (−0.572 + 1.76i)4-s + (−1.30 − 0.951i)5-s + (−0.309 − 0.224i)6-s + (−0.309 + 0.951i)7-s + (0.454 + 1.40i)8-s + (−0.809 + 0.587i)9-s − 0.618·10-s + (−2.19 − 2.48i)11-s + 1.85·12-s + (3.42 − 2.48i)13-s + (0.118 + 0.363i)14-s + (−0.499 + 1.53i)15-s + (−2.54 − 1.84i)16-s + (6.35 + 4.61i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.158i)2-s + (−0.178 − 0.549i)3-s + (−0.286 + 0.881i)4-s + (−0.585 − 0.425i)5-s + (−0.126 − 0.0916i)6-s + (−0.116 + 0.359i)7-s + (0.160 + 0.495i)8-s + (−0.269 + 0.195i)9-s − 0.195·10-s + (−0.660 − 0.750i)11-s + 0.535·12-s + (0.950 − 0.690i)13-s + (0.0315 + 0.0970i)14-s + (−0.129 + 0.397i)15-s + (−0.636 − 0.462i)16-s + (1.54 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.970 + 0.242i$
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: \((2,\ 33,\ (\ :1/2),\ 0.970 + 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.685332 - 0.0842454i\)
\(L(\frac12)\) \(\approx\) \(0.685332 - 0.0842454i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 + 2.48i)T \)
good2 \( 1 + (-0.309 + 0.224i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (1.30 + 0.951i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.309 - 0.951i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.42 + 2.48i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-6.35 - 4.61i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.263 + 0.812i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.23T + 23T^{2} \)
29 \( 1 + (1.85 - 5.70i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.11 - 2.99i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.545 - 1.67i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.30 + 4.02i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.70T + 43T^{2} \)
47 \( 1 + (-0.336 - 1.03i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.11 + 1.53i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.97 + 9.14i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.92 + 5.03i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 4.85T + 67T^{2} \)
71 \( 1 + (-4.30 - 3.13i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.38 + 7.33i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.89 - 6.46i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-6.04 - 4.39i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 3.76T + 89T^{2} \)
97 \( 1 + (0.927 - 0.673i)T + (29.9 - 92.2i)T^{2} \)
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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

−16.69140000180274586172293298347, −15.82376536896543676893985401643, −14.05339806885900157386526069329, −12.80408060483079045697412295504, −12.22878587209579477609451956978, −10.81741248565105706753367219342, −8.553772017013687259261755834245, −7.82752521761884850187564116000, −5.65105796164047322365801015824, −3.50615854187679487427044393002, 4.07106803246905813455387311145, 5.70602665208127893703918281374, 7.44629319321667452396416432202, 9.496414475514369998389671803402, 10.48627858732892218675964487356, 11.73748930687682857742956772434, 13.52849772848038552864046426470, 14.59678019166787755615263599236, 15.57961920766766971037884281083, 16.48660660504225864507154828962

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