Properties

Label 2-33-33.2-c3-0-3
Degree $2$
Conductor $33$
Sign $0.970 - 0.239i$
Analytic cond. $1.94706$
Root an. cond. $1.39537$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.448 − 1.37i)2-s + (−2.40 + 4.60i)3-s + (4.76 + 3.46i)4-s + (14.6 − 4.75i)5-s + (5.27 + 5.38i)6-s + (−7.94 + 10.9i)7-s + (16.3 − 11.8i)8-s + (−15.4 − 22.1i)9-s − 22.3i·10-s + (−36.2 + 4.26i)11-s + (−27.4 + 13.6i)12-s + (21.5 + 6.99i)13-s + (11.5 + 15.8i)14-s + (−13.3 + 78.8i)15-s + (5.53 + 17.0i)16-s + (−27.3 − 84.0i)17-s + ⋯
L(s)  = 1  + (0.158 − 0.487i)2-s + (−0.463 + 0.886i)3-s + (0.596 + 0.433i)4-s + (1.30 − 0.425i)5-s + (0.358 + 0.366i)6-s + (−0.428 + 0.590i)7-s + (0.720 − 0.523i)8-s + (−0.570 − 0.821i)9-s − 0.705i·10-s + (−0.993 + 0.116i)11-s + (−0.660 + 0.327i)12-s + (0.459 + 0.149i)13-s + (0.219 + 0.302i)14-s + (−0.229 + 1.35i)15-s + (0.0864 + 0.266i)16-s + (−0.389 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.970 - 0.239i$
Analytic conductor: \(1.94706\)
Root analytic conductor: \(1.39537\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :3/2),\ 0.970 - 0.239i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.44284 + 0.175185i\)
\(L(\frac12)\) \(\approx\) \(1.44284 + 0.175185i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.40 - 4.60i)T \)
11 \( 1 + (36.2 - 4.26i)T \)
good2 \( 1 + (-0.448 + 1.37i)T + (-6.47 - 4.70i)T^{2} \)
5 \( 1 + (-14.6 + 4.75i)T + (101. - 73.4i)T^{2} \)
7 \( 1 + (7.94 - 10.9i)T + (-105. - 326. i)T^{2} \)
13 \( 1 + (-21.5 - 6.99i)T + (1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (27.3 + 84.0i)T + (-3.97e3 + 2.88e3i)T^{2} \)
19 \( 1 + (60.6 + 83.4i)T + (-2.11e3 + 6.52e3i)T^{2} \)
23 \( 1 + 42.7iT - 1.21e4T^{2} \)
29 \( 1 + (6.11 + 4.44i)T + (7.53e3 + 2.31e4i)T^{2} \)
31 \( 1 + (67.5 - 207. i)T + (-2.41e4 - 1.75e4i)T^{2} \)
37 \( 1 + (-251. - 182. i)T + (1.56e4 + 4.81e4i)T^{2} \)
41 \( 1 + (268. - 195. i)T + (2.12e4 - 6.55e4i)T^{2} \)
43 \( 1 + 419. iT - 7.95e4T^{2} \)
47 \( 1 + (-100. - 137. i)T + (-3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (329. + 107. i)T + (1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (30.0 - 41.3i)T + (-6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (-52.8 + 17.1i)T + (1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 + 5.78T + 3.00e5T^{2} \)
71 \( 1 + (-69.3 + 22.5i)T + (2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (170. - 234. i)T + (-1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (-605. - 196. i)T + (3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (-360. - 1.10e3i)T + (-4.62e5 + 3.36e5i)T^{2} \)
89 \( 1 + 1.31e3iT - 7.04e5T^{2} \)
97 \( 1 + (-189. + 582. i)T + (-7.38e5 - 5.36e5i)T^{2} \)
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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

−16.24236310173120116955875927205, −15.44428870784239953835672261516, −13.56501852693263076937046910450, −12.55910827457646280707570182935, −11.19962992622101755945287175838, −10.10906896026502446824730120374, −8.941968250581134484834657117684, −6.47039966637703279010069717384, −4.99306458742373020582220479335, −2.68192795747059568621103052442, 2.00069722674876258151912095171, 5.75464803918418525169470857451, 6.39504536318966823536740720606, 7.82495089063352860381302678718, 10.22021944907876940177758545044, 10.95782273535726039926411257075, 12.93934023458807832408488287091, 13.66823590622608904187092929467, 14.86304720950687114502045607875, 16.41138219313471367941141294384

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