Properties

Label 2-33-33.2-c1-0-0
Degree $2$
Conductor $33$
Sign $0.00494 - 0.999i$
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  + (−0.587 + 1.80i)2-s + (−1.67 + 0.451i)3-s + (−1.30 − 0.951i)4-s + (2.48 − 0.809i)5-s + (0.166 − 3.29i)6-s + (0.427 − 0.587i)7-s + (−0.587 + 0.427i)8-s + (2.59 − 1.50i)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 + (−3.79 + 2.47i)15-s + (−1.42 − 4.39i)16-s + (0.812 + 2.5i)17-s + ⋯
L(s)  = 1  + (−0.415 + 1.27i)2-s + (−0.965 + 0.260i)3-s + (−0.654 − 0.475i)4-s + (1.11 − 0.361i)5-s + (0.0681 − 1.34i)6-s + (0.161 − 0.222i)7-s + (−0.207 + 0.150i)8-s + (0.864 − 0.502i)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 + (−0.980 + 0.639i)15-s + (−0.356 − 1.09i)16-s + (0.197 + 0.606i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00494 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00494 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.00494 - 0.999i$
Analytic conductor: \(0.263506\)
Root analytic conductor: \(0.513328\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1/2),\ 0.00494 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.383111 + 0.381221i\)
\(L(\frac12)\) \(\approx\) \(0.383111 + 0.381221i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 - 0.451i)T \)
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

−16.97870044362324053534021418496, −16.29710081469198663025228156273, −15.07334365408290441942933458137, −13.74117245902369819289835941464, −12.27658613770815094223420227829, −10.55567719834630769600427875861, −9.369057630643368572885790977418, −7.71151005689040837509595628476, −6.10648580164832432242827228769, −5.29487782968491994744339627738, 2.19885899916062324671191428906, 5.28318210902482122110674874767, 7.01011624687125594327007714485, 9.537700291041815628266509310609, 10.27681012917364705659367509047, 11.46347536066987582343912582302, 12.46652927391848171765887401288, 13.60075146461187558908129571292, 15.39262604941124465722338644609, 17.06324554972991059661586008793

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