Properties

Label 2-33-11.5-c3-0-1
Degree $2$
Conductor $33$
Sign $-0.453 - 0.891i$
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  + (1.58 + 4.87i)2-s + (2.42 + 1.76i)3-s + (−14.7 + 10.7i)4-s + (4.61 − 14.2i)5-s + (−4.75 + 14.6i)6-s + (9.98 − 7.25i)7-s + (−42.5 − 30.9i)8-s + (2.78 + 8.55i)9-s + 76.5·10-s + (−31.7 + 18.0i)11-s − 54.7·12-s + (9.43 + 29.0i)13-s + (51.1 + 37.1i)14-s + (36.2 − 26.3i)15-s + (38.1 − 117. i)16-s + (40.8 − 125. i)17-s + ⋯
L(s)  = 1  + (0.559 + 1.72i)2-s + (0.467 + 0.339i)3-s + (−1.84 + 1.34i)4-s + (0.412 − 1.27i)5-s + (−0.323 + 0.994i)6-s + (0.539 − 0.391i)7-s + (−1.88 − 1.36i)8-s + (0.103 + 0.317i)9-s + 2.42·10-s + (−0.869 + 0.494i)11-s − 1.31·12-s + (0.201 + 0.619i)13-s + (0.976 + 0.709i)14-s + (0.624 − 0.453i)15-s + (0.595 − 1.83i)16-s + (0.582 − 1.79i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.453 - 0.891i$
Analytic conductor: \(1.94706\)
Root analytic conductor: \(1.39537\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :3/2),\ -0.453 - 0.891i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.897738 + 1.46450i\)
\(L(\frac12)\) \(\approx\) \(0.897738 + 1.46450i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.42 - 1.76i)T \)
11 \( 1 + (31.7 - 18.0i)T \)
good2 \( 1 + (-1.58 - 4.87i)T + (-6.47 + 4.70i)T^{2} \)
5 \( 1 + (-4.61 + 14.2i)T + (-101. - 73.4i)T^{2} \)
7 \( 1 + (-9.98 + 7.25i)T + (105. - 326. i)T^{2} \)
13 \( 1 + (-9.43 - 29.0i)T + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (-40.8 + 125. i)T + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (-18.0 - 13.1i)T + (2.11e3 + 6.52e3i)T^{2} \)
23 \( 1 + 158.T + 1.21e4T^{2} \)
29 \( 1 + (-38.6 + 28.0i)T + (7.53e3 - 2.31e4i)T^{2} \)
31 \( 1 + (-21.0 - 64.7i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (128. - 93.1i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (231. + 168. i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 - 103.T + 7.95e4T^{2} \)
47 \( 1 + (-375. - 272. i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (1.86 + 5.75i)T + (-1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (-179. + 130. i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (84.7 - 260. i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 - 187.T + 3.00e5T^{2} \)
71 \( 1 + (141. - 435. i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (-218. + 158. i)T + (1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (-152. - 469. i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (83.7 - 257. i)T + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 + 77.4T + 7.04e5T^{2} \)
97 \( 1 + (392. + 1.20e3i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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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

−16.25862459859500339924867242022, −15.69701574327972817146149569626, −14.12227617255913400839609721884, −13.67845793360088665825971854789, −12.30870843348220573635669900596, −9.661406609950008453603059246715, −8.469332684521712009143578890006, −7.39851912532690530534118277502, −5.39077542762661344446445747360, −4.48069246512469574362811245238, 2.13414026844156928401540389062, 3.45564765013432837834607068994, 5.77708058162761477179307767467, 8.248003880112029675886192780825, 10.13659657142779914932870289921, 10.78463097150437898347751349656, 12.13226392462098465682536730995, 13.31395404989300196071006021160, 14.23462064454013541878360773112, 15.13729161605475533316380032392

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