Properties

Label 2-33-11.5-c1-0-1
Degree $2$
Conductor $33$
Sign $0.0219 + 0.999i$
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  + (−0.809 − 2.48i)2-s + (0.809 + 0.587i)3-s + (−3.92 + 2.85i)4-s + (−0.190 + 0.587i)5-s + (0.809 − 2.48i)6-s + (0.809 − 0.587i)7-s + (6.04 + 4.39i)8-s + (0.309 + 0.951i)9-s + 1.61·10-s + (−3.30 + 0.224i)11-s − 4.85·12-s + (0.0729 + 0.224i)13-s + (−2.11 − 1.53i)14-s + (−0.5 + 0.363i)15-s + (3.04 − 9.37i)16-s + (−0.354 + 1.08i)17-s + ⋯
L(s)  = 1  + (−0.572 − 1.76i)2-s + (0.467 + 0.339i)3-s + (−1.96 + 1.42i)4-s + (−0.0854 + 0.262i)5-s + (0.330 − 1.01i)6-s + (0.305 − 0.222i)7-s + (2.13 + 1.55i)8-s + (0.103 + 0.317i)9-s + 0.511·10-s + (−0.997 + 0.0676i)11-s − 1.40·12-s + (0.0202 + 0.0622i)13-s + (−0.566 − 0.411i)14-s + (−0.129 + 0.0937i)15-s + (0.761 − 2.34i)16-s + (−0.0858 + 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.0219 + 0.999i$
Analytic conductor: \(0.263506\)
Root analytic conductor: \(0.513328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1/2),\ 0.0219 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.417216 - 0.408142i\)
\(L(\frac12)\) \(\approx\) \(0.417216 - 0.408142i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (3.30 - 0.224i)T \)
good2 \( 1 + (0.809 + 2.48i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (0.190 - 0.587i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-0.809 + 0.587i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.0729 - 0.224i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.354 - 1.08i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.73 + 3.44i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 0.236T + 23T^{2} \)
29 \( 1 + (-4.85 + 3.52i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.88 + 5.79i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.04 + 3.66i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.190 + 0.138i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 6.70T + 43T^{2} \)
47 \( 1 + (-8.16 - 5.93i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.118 + 0.363i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (5.97 - 4.33i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.57 - 10.9i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 1.85T + 67T^{2} \)
71 \( 1 + (-3.19 + 9.82i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.61 + 3.35i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.39 - 10.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.454 + 1.40i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 8.23T + 89T^{2} \)
97 \( 1 + (-2.42 - 7.46i)T + (-78.4 + 57.0i)T^{2} \)
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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

−16.95132035947679451569949978950, −15.17156313286988885866690848012, −13.63358303594250610660446094091, −12.68301172593896206414019449094, −11.14766597732641612746030580536, −10.43942286907456188046405755049, −9.148644582998023905922649744957, −7.923647101006141623159654241072, −4.39487915900050490447519620477, −2.62604570344948288392588188459, 5.02455153538065297246012103544, 6.61462173149129510325974246137, 8.012237238379769935991718337682, 8.727398947388126533120623200704, 10.29314469740978559945928134412, 12.71538019834014812581411881352, 13.98210287115459103407761090817, 14.94034050981654425889176507722, 15.91332035097733474512034777870, 16.91882142420956781373832330125

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