Properties

Label 2-33-11.6-c4-0-3
Degree $2$
Conductor $33$
Sign $0.951 - 0.307i$
Analytic cond. $3.41120$
Root an. cond. $1.84694$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.49 + 1.45i)2-s + (−4.20 − 3.05i)3-s + (5.11 − 3.71i)4-s + (5.35 − 16.4i)5-s + (23.3 + 7.58i)6-s + (48.2 + 66.3i)7-s + (26.8 − 37.0i)8-s + (8.34 + 25.6i)9-s + 81.9i·10-s + (120. + 2.78i)11-s − 32.8·12-s + (162. − 52.6i)13-s + (−313. − 227. i)14-s + (−72.8 + 52.9i)15-s + (−98.0 + 301. i)16-s + (−314. − 102. i)17-s + ⋯
L(s)  = 1  + (−1.12 + 0.364i)2-s + (−0.467 − 0.339i)3-s + (0.319 − 0.232i)4-s + (0.214 − 0.659i)5-s + (0.648 + 0.210i)6-s + (0.984 + 1.35i)7-s + (0.420 − 0.578i)8-s + (0.103 + 0.317i)9-s + 0.819i·10-s + (0.999 + 0.0230i)11-s − 0.227·12-s + (0.959 − 0.311i)13-s + (−1.60 − 1.16i)14-s + (−0.323 + 0.235i)15-s + (−0.382 + 1.17i)16-s + (−1.08 − 0.353i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.951 - 0.307i$
Analytic conductor: \(3.41120\)
Root analytic conductor: \(1.84694\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :2),\ 0.951 - 0.307i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.784385 + 0.123804i\)
\(L(\frac12)\) \(\approx\) \(0.784385 + 0.123804i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.20 + 3.05i)T \)
11 \( 1 + (-120. - 2.78i)T \)
good2 \( 1 + (4.49 - 1.45i)T + (12.9 - 9.40i)T^{2} \)
5 \( 1 + (-5.35 + 16.4i)T + (-505. - 367. i)T^{2} \)
7 \( 1 + (-48.2 - 66.3i)T + (-741. + 2.28e3i)T^{2} \)
13 \( 1 + (-162. + 52.6i)T + (2.31e4 - 1.67e4i)T^{2} \)
17 \( 1 + (314. + 102. i)T + (6.75e4 + 4.90e4i)T^{2} \)
19 \( 1 + (-159. + 219. i)T + (-4.02e4 - 1.23e5i)T^{2} \)
23 \( 1 - 414.T + 2.79e5T^{2} \)
29 \( 1 + (-631. - 868. i)T + (-2.18e5 + 6.72e5i)T^{2} \)
31 \( 1 + (-167. - 516. i)T + (-7.47e5 + 5.42e5i)T^{2} \)
37 \( 1 + (508. - 369. i)T + (5.79e5 - 1.78e6i)T^{2} \)
41 \( 1 + (144. - 198. i)T + (-8.73e5 - 2.68e6i)T^{2} \)
43 \( 1 + 3.52e3iT - 3.41e6T^{2} \)
47 \( 1 + (1.90e3 + 1.38e3i)T + (1.50e6 + 4.64e6i)T^{2} \)
53 \( 1 + (-316. - 973. i)T + (-6.38e6 + 4.63e6i)T^{2} \)
59 \( 1 + (1.75e3 - 1.27e3i)T + (3.74e6 - 1.15e7i)T^{2} \)
61 \( 1 + (-3.67e3 - 1.19e3i)T + (1.12e7 + 8.13e6i)T^{2} \)
67 \( 1 + 70.0T + 2.01e7T^{2} \)
71 \( 1 + (141. - 435. i)T + (-2.05e7 - 1.49e7i)T^{2} \)
73 \( 1 + (3.91e3 + 5.38e3i)T + (-8.77e6 + 2.70e7i)T^{2} \)
79 \( 1 + (-7.78e3 + 2.52e3i)T + (3.15e7 - 2.28e7i)T^{2} \)
83 \( 1 + (-301. - 98.1i)T + (3.83e7 + 2.78e7i)T^{2} \)
89 \( 1 + 5.30e3T + 6.27e7T^{2} \)
97 \( 1 + (1.62e3 + 5.00e3i)T + (-7.16e7 + 5.20e7i)T^{2} \)
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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

−16.31369804719278110062457846970, −15.25700513410845911216684943023, −13.48387308269842707189971480089, −12.15247836883124457066365738173, −10.96948149142896441870002279276, −8.985924625887846365185142385125, −8.617506048004414928093891714084, −6.79149516157012544527593153888, −5.06955638051269447172872679153, −1.31499547195132819186958753782, 1.22666303533443917939449221089, 4.34178479717416912650465240982, 6.67071322382918047098921856573, 8.251781887550054190870764096536, 9.728094151679720436886732553641, 10.89058780498468468243664746822, 11.33007371587206851693159099326, 13.71245155934525230703499835352, 14.61098188511678534006277867902, 16.42194333986839336440328690523

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