Properties

Label 2-33-11.8-c2-0-0
Degree $2$
Conductor $33$
Sign $0.999 - 0.0174i$
Analytic cond. $0.899184$
Root an. cond. $0.948253$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.184 − 0.253i)2-s + (0.535 + 1.64i)3-s + (1.20 − 3.71i)4-s + (5.99 + 4.35i)5-s + (0.318 − 0.438i)6-s + (−9.53 − 3.09i)7-s + (−2.35 + 0.764i)8-s + (−2.42 + 1.76i)9-s − 2.32i·10-s + (−10.2 − 3.93i)11-s + 6.75·12-s + (2.00 + 2.75i)13-s + (0.970 + 2.98i)14-s + (−3.96 + 12.2i)15-s + (−12.0 − 8.71i)16-s + (−9.14 + 12.5i)17-s + ⋯
L(s)  = 1  + (−0.0920 − 0.126i)2-s + (0.178 + 0.549i)3-s + (0.301 − 0.927i)4-s + (1.19 + 0.871i)5-s + (0.0531 − 0.0731i)6-s + (−1.36 − 0.442i)7-s + (−0.294 + 0.0955i)8-s + (−0.269 + 0.195i)9-s − 0.232i·10-s + (−0.933 − 0.358i)11-s + 0.563·12-s + (0.153 + 0.211i)13-s + (0.0692 + 0.213i)14-s + (−0.264 + 0.814i)15-s + (−0.750 − 0.544i)16-s + (−0.537 + 0.740i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0174i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.999 - 0.0174i$
Analytic conductor: \(0.899184\)
Root analytic conductor: \(0.948253\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1),\ 0.999 - 0.0174i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.07879 + 0.00938772i\)
\(L(\frac12)\) \(\approx\) \(1.07879 + 0.00938772i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.535 - 1.64i)T \)
11 \( 1 + (10.2 + 3.93i)T \)
good2 \( 1 + (0.184 + 0.253i)T + (-1.23 + 3.80i)T^{2} \)
5 \( 1 + (-5.99 - 4.35i)T + (7.72 + 23.7i)T^{2} \)
7 \( 1 + (9.53 + 3.09i)T + (39.6 + 28.8i)T^{2} \)
13 \( 1 + (-2.00 - 2.75i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (9.14 - 12.5i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (-29.1 + 9.46i)T + (292. - 212. i)T^{2} \)
23 \( 1 - 7.67T + 529T^{2} \)
29 \( 1 + (-3.21 - 1.04i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (-3.27 + 2.37i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (0.734 - 2.25i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-6.69 + 2.17i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 - 3.99iT - 1.84e3T^{2} \)
47 \( 1 + (-15.2 - 47.0i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-48.3 + 35.1i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (3.39 - 10.4i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (-43.6 + 60.1i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 + 3.22T + 4.48e3T^{2} \)
71 \( 1 + (94.5 + 68.7i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-17.8 - 5.78i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (2.06 + 2.84i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (86.4 - 119. i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 + 65.8T + 7.92e3T^{2} \)
97 \( 1 + (52.4 - 38.1i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−16.27617110536437385927014647174, −15.37130443916638479461556309146, −14.06444709357203262073258886312, −13.29455306748765307806619509521, −10.98034822453394853065957318825, −10.14617726852584079758915368421, −9.414849626623867607636312266442, −6.77404040014754064869687156632, −5.65463462597109520945196575610, −2.85141994790067530153574085742, 2.75245676862392108387586191427, 5.66684548670637219790598277779, 7.16882412832442390930857642734, 8.782208693993455986944533521542, 9.820611425739183496023579811053, 12.05088044442935858022597532266, 13.00645820561289224175521406267, 13.53454988467521956418465371740, 15.73612441986101390246673765710, 16.48439531518953433049746996693

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