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Properties

Label 2-33-33.17-c1-0-0
Degree $2$
Conductor $33$
Sign $0.662 - 0.748i$
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  + (0.587 + 1.80i)2-s + (−0.945 − 1.45i)3-s + (−1.30 + 0.951i)4-s + (−2.48 − 0.809i)5-s + (2.06 − 2.56i)6-s + (0.427 + 0.587i)7-s + (0.587 + 0.427i)8-s + (−1.21 + 2.74i)9-s − 4.97i·10-s + (2.12 − 2.54i)11-s + (2.61 + i)12-s + (−2.92 + 0.951i)13-s + (−0.812 + 1.11i)14-s + (1.18 + 4.37i)15-s + (−1.42 + 4.39i)16-s + (−0.812 + 2.5i)17-s + ⋯
L(s)  = 1  + (0.415 + 1.27i)2-s + (−0.546 − 0.837i)3-s + (−0.654 + 0.475i)4-s + (−1.11 − 0.361i)5-s + (0.844 − 1.04i)6-s + (0.161 + 0.222i)7-s + (0.207 + 0.150i)8-s + (−0.403 + 0.914i)9-s − 1.57i·10-s + (0.641 − 0.767i)11-s + (0.755 + 0.288i)12-s + (−0.811 + 0.263i)13-s + (−0.217 + 0.298i)14-s + (0.304 + 1.13i)15-s + (−0.356 + 1.09i)16-s + (−0.197 + 0.606i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.662 - 0.748i$
Analytic conductor: \(0.263506\)
Root analytic conductor: \(0.513328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1/2),\ 0.662 - 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.646360 + 0.290981i\)
\(L(\frac12)\) \(\approx\) \(0.646360 + 0.290981i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.945 + 1.45i)T \)
11 \( 1 + (-2.12 + 2.54i)T \)
good2 \( 1 + (-0.587 - 1.80i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (2.48 + 0.809i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.427 - 0.587i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (2.92 - 0.951i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.812 - 2.5i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.5 + 3.44i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 1.76iT - 23T^{2} \)
29 \( 1 + (-3.07 + 2.23i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.263 + 0.812i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.42 - 1.76i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.48 - 1.80i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 1.62iT - 43T^{2} \)
47 \( 1 + (4.30 - 5.92i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.61 - 1.5i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.53 - 2.11i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.04 + 1.31i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 8.32T + 67T^{2} \)
71 \( 1 + (9.82 + 3.19i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-8.94 - 12.3i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-10.1 + 3.30i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.47 + 13.7i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 9.47iT - 89T^{2} \)
97 \( 1 + (2.04 + 6.29i)T + (-78.4 + 57.0i)T^{2} \)
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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

−16.72479950346996863300548924735, −15.89602961035261059001988887083, −14.70485910759990035812733311732, −13.54590225893332775354832696624, −12.17844678989521950832237461898, −11.20735018750319948846883652667, −8.496570526395388735924808322955, −7.47914318088567035951254171921, −6.26580821165782136539791773101, −4.72388620998229816799884347335, 3.51129869137378542111995479933, 4.72367441207294332970852923949, 7.30187987952647945022884127068, 9.602650744913680615819656121831, 10.68565297511309392496008733754, 11.75160345701233559367605905686, 12.29315084903359854766200797307, 14.24377121547897808558386077673, 15.39104017694440349294046284039, 16.52305634220327866421831512312

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