Properties

Label 2-33-33.2-c5-0-0
Degree $2$
Conductor $33$
Sign $0.471 + 0.881i$
Analytic cond. $5.29266$
Root an. cond. $2.30057$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.08 + 9.48i)2-s + (−12.5 + 9.26i)3-s + (−54.5 − 39.6i)4-s + (16.3 − 5.32i)5-s + (−49.2 − 147. i)6-s + (−74.1 + 102. i)7-s + (286. − 207. i)8-s + (71.2 − 232. i)9-s + 171. i·10-s + (293. + 274. i)11-s + (1.05e3 − 8.74i)12-s + (−289. − 94.2i)13-s + (−739. − 1.01e3i)14-s + (−156. + 218. i)15-s + (422. + 1.30e3i)16-s + (−415. − 1.27e3i)17-s + ⋯
L(s)  = 1  + (−0.544 + 1.67i)2-s + (−0.804 + 0.594i)3-s + (−1.70 − 1.23i)4-s + (0.293 − 0.0952i)5-s + (−0.558 − 1.67i)6-s + (−0.572 + 0.787i)7-s + (1.58 − 1.14i)8-s + (0.293 − 0.956i)9-s + 0.543i·10-s + (0.730 + 0.682i)11-s + (2.10 − 0.0175i)12-s + (−0.475 − 0.154i)13-s + (−1.00 − 1.38i)14-s + (−0.179 + 0.250i)15-s + (0.412 + 1.27i)16-s + (−0.349 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.471 + 0.881i$
Analytic conductor: \(5.29266\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ 0.471 + 0.881i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.142111 - 0.0851465i\)
\(L(\frac12)\) \(\approx\) \(0.142111 - 0.0851465i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (12.5 - 9.26i)T \)
11 \( 1 + (-293. - 274. i)T \)
good2 \( 1 + (3.08 - 9.48i)T + (-25.8 - 18.8i)T^{2} \)
5 \( 1 + (-16.3 + 5.32i)T + (2.52e3 - 1.83e3i)T^{2} \)
7 \( 1 + (74.1 - 102. i)T + (-5.19e3 - 1.59e4i)T^{2} \)
13 \( 1 + (289. + 94.2i)T + (3.00e5 + 2.18e5i)T^{2} \)
17 \( 1 + (415. + 1.27e3i)T + (-1.14e6 + 8.34e5i)T^{2} \)
19 \( 1 + (1.59e3 + 2.19e3i)T + (-7.65e5 + 2.35e6i)T^{2} \)
23 \( 1 - 3.56e3iT - 6.43e6T^{2} \)
29 \( 1 + (3.06e3 + 2.23e3i)T + (6.33e6 + 1.95e7i)T^{2} \)
31 \( 1 + (-848. + 2.61e3i)T + (-2.31e7 - 1.68e7i)T^{2} \)
37 \( 1 + (159. + 116. i)T + (2.14e7 + 6.59e7i)T^{2} \)
41 \( 1 + (1.36e4 - 9.89e3i)T + (3.58e7 - 1.10e8i)T^{2} \)
43 \( 1 - 6.55e3iT - 1.47e8T^{2} \)
47 \( 1 + (6.78e3 + 9.33e3i)T + (-7.08e7 + 2.18e8i)T^{2} \)
53 \( 1 + (-1.24e4 - 4.05e3i)T + (3.38e8 + 2.45e8i)T^{2} \)
59 \( 1 + (1.56e4 - 2.15e4i)T + (-2.20e8 - 6.79e8i)T^{2} \)
61 \( 1 + (2.71e4 - 8.82e3i)T + (6.83e8 - 4.96e8i)T^{2} \)
67 \( 1 + 2.85e4T + 1.35e9T^{2} \)
71 \( 1 + (-6.58e4 + 2.14e4i)T + (1.45e9 - 1.06e9i)T^{2} \)
73 \( 1 + (-1.83e4 + 2.53e4i)T + (-6.40e8 - 1.97e9i)T^{2} \)
79 \( 1 + (8.08e4 + 2.62e4i)T + (2.48e9 + 1.80e9i)T^{2} \)
83 \( 1 + (-2.29e4 - 7.06e4i)T + (-3.18e9 + 2.31e9i)T^{2} \)
89 \( 1 + 4.92e4iT - 5.58e9T^{2} \)
97 \( 1 + (7.33e3 - 2.25e4i)T + (-6.94e9 - 5.04e9i)T^{2} \)
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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

−16.74086805906629468502557823957, −15.39774867571659264502863414053, −15.17979183975570808616923987747, −13.37226479567040480470068915029, −11.68483714910413032942812425170, −9.653661346074092742784604890837, −9.184119082663599312843314661799, −7.12516048606192063433533786092, −6.02771870912570195263486890485, −4.82074237563079701148253840808, 0.13144675794398368147209577705, 1.78325463446714374316265036675, 3.95953023841122696152151378399, 6.43185536790755833954972402296, 8.428863172371675614592618986595, 10.14967412805546156654395219143, 10.77871800914428172740390693004, 12.12354773583956602592702865208, 12.88270771639955937395585532699, 14.06383191546334119514090986943

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