Properties

Label 2-33-33.17-c3-0-9
Degree $2$
Conductor $33$
Sign $-0.506 - 0.862i$
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  + (−1.66 − 5.13i)2-s + (−2.33 + 4.64i)3-s + (−17.1 + 12.4i)4-s + (−7.03 − 2.28i)5-s + (27.7 + 4.22i)6-s + (−8.32 − 11.4i)7-s + (57.6 + 41.8i)8-s + (−16.1 − 21.6i)9-s + 39.9i·10-s + (−36.4 + 1.15i)11-s + (−17.8 − 108. i)12-s + (−5.34 + 1.73i)13-s + (−44.9 + 61.9i)14-s + (26.9 − 27.3i)15-s + (66.6 − 204. i)16-s + (16.4 − 50.7i)17-s + ⋯
L(s)  = 1  + (−0.590 − 1.81i)2-s + (−0.448 + 0.893i)3-s + (−2.14 + 1.55i)4-s + (−0.628 − 0.204i)5-s + (1.88 + 0.287i)6-s + (−0.449 − 0.618i)7-s + (2.54 + 1.85i)8-s + (−0.597 − 0.801i)9-s + 1.26i·10-s + (−0.999 + 0.0316i)11-s + (−0.429 − 2.61i)12-s + (−0.113 + 0.0370i)13-s + (−0.858 + 1.18i)14-s + (0.464 − 0.470i)15-s + (1.04 − 3.20i)16-s + (0.235 − 0.724i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.506 - 0.862i$
Analytic conductor: \(1.94706\)
Root analytic conductor: \(1.39537\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :3/2),\ -0.506 - 0.862i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0635300 + 0.110967i\)
\(L(\frac12)\) \(\approx\) \(0.0635300 + 0.110967i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.33 - 4.64i)T \)
11 \( 1 + (36.4 - 1.15i)T \)
good2 \( 1 + (1.66 + 5.13i)T + (-6.47 + 4.70i)T^{2} \)
5 \( 1 + (7.03 + 2.28i)T + (101. + 73.4i)T^{2} \)
7 \( 1 + (8.32 + 11.4i)T + (-105. + 326. i)T^{2} \)
13 \( 1 + (5.34 - 1.73i)T + (1.77e3 - 1.29e3i)T^{2} \)
17 \( 1 + (-16.4 + 50.7i)T + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (20.6 - 28.4i)T + (-2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 - 64.1iT - 1.21e4T^{2} \)
29 \( 1 + (-113. + 82.7i)T + (7.53e3 - 2.31e4i)T^{2} \)
31 \( 1 + (9.10 + 28.0i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (171. - 124. i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (213. + 154. i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 - 385. iT - 7.95e4T^{2} \)
47 \( 1 + (133. - 184. i)T + (-3.20e4 - 9.87e4i)T^{2} \)
53 \( 1 + (185. - 60.3i)T + (1.20e5 - 8.75e4i)T^{2} \)
59 \( 1 + (277. + 381. i)T + (-6.34e4 + 1.95e5i)T^{2} \)
61 \( 1 + (616. + 200. i)T + (1.83e5 + 1.33e5i)T^{2} \)
67 \( 1 - 823.T + 3.00e5T^{2} \)
71 \( 1 + (605. + 196. i)T + (2.89e5 + 2.10e5i)T^{2} \)
73 \( 1 + (-110. - 152. i)T + (-1.20e5 + 3.69e5i)T^{2} \)
79 \( 1 + (-816. + 265. i)T + (3.98e5 - 2.89e5i)T^{2} \)
83 \( 1 + (-67.7 + 208. i)T + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 + 217. iT - 7.04e5T^{2} \)
97 \( 1 + (537. + 1.65e3i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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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

−15.78342256371999102310315179802, −13.75907895767824164724367519809, −12.42860853478052352325687088681, −11.45635886415112434371157645258, −10.39875332445655324708699657774, −9.598464690110522423787721588700, −8.071798025618221242739724879443, −4.67005612424613269613916905744, −3.30898920005320025250843532531, −0.14515877396324156328585141957, 5.29788502166139771433919241820, 6.53042038287895966630924149541, 7.67232409657361899556210538583, 8.662257448563688394314268561958, 10.49257608625194179626167907966, 12.49187294997457263276950754843, 13.69494976779638517364672242600, 15.06993436016694079046216552586, 15.90506317881288574914296818753, 16.94825407771734689743871163229

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