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Properties

Label 2-33-33.32-c5-0-3
Degree $2$
Conductor $33$
Sign $-0.484 - 0.874i$
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  + 3.58·2-s + (0.133 + 15.5i)3-s − 19.1·4-s + 11.8i·5-s + (0.480 + 55.9i)6-s + 150. i·7-s − 183.·8-s + (−242. + 4.17i)9-s + 42.6i·10-s + (352. − 191. i)11-s + (−2.56 − 297. i)12-s + 778. i·13-s + 541. i·14-s + (−185. + 1.59i)15-s − 47.1·16-s + 1.25e3·17-s + ⋯
L(s)  = 1  + 0.634·2-s + (0.00859 + 0.999i)3-s − 0.597·4-s + 0.212i·5-s + (0.00545 + 0.634i)6-s + 1.16i·7-s − 1.01·8-s + (−0.999 + 0.0171i)9-s + 0.134i·10-s + (0.878 − 0.477i)11-s + (−0.00513 − 0.597i)12-s + 1.27i·13-s + 0.738i·14-s + (−0.212 + 0.00182i)15-s − 0.0460·16-s + 1.05·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.484 - 0.874i$
Analytic conductor: \(5.29266\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ -0.484 - 0.874i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.786381 + 1.33523i\)
\(L(\frac12)\) \(\approx\) \(0.786381 + 1.33523i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.133 - 15.5i)T \)
11 \( 1 + (-352. + 191. i)T \)
good2 \( 1 - 3.58T + 32T^{2} \)
5 \( 1 - 11.8iT - 3.12e3T^{2} \)
7 \( 1 - 150. iT - 1.68e4T^{2} \)
13 \( 1 - 778. iT - 3.71e5T^{2} \)
17 \( 1 - 1.25e3T + 1.41e6T^{2} \)
19 \( 1 + 1.21e3iT - 2.47e6T^{2} \)
23 \( 1 - 880. iT - 6.43e6T^{2} \)
29 \( 1 - 3.20e3T + 2.05e7T^{2} \)
31 \( 1 + 6.44e3T + 2.86e7T^{2} \)
37 \( 1 + 9.31e3T + 6.93e7T^{2} \)
41 \( 1 - 5.39e3T + 1.15e8T^{2} \)
43 \( 1 - 2.08e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.12e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.39e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.29e4iT - 7.14e8T^{2} \)
61 \( 1 + 2.72e4iT - 8.44e8T^{2} \)
67 \( 1 - 2.02e4T + 1.35e9T^{2} \)
71 \( 1 + 5.64e4iT - 1.80e9T^{2} \)
73 \( 1 + 1.20e4iT - 2.07e9T^{2} \)
79 \( 1 + 1.10e4iT - 3.07e9T^{2} \)
83 \( 1 + 5.15e4T + 3.93e9T^{2} \)
89 \( 1 + 8.41e4iT - 5.58e9T^{2} \)
97 \( 1 - 9.93e4T + 8.58e9T^{2} \)
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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

−15.86344447432515370667338385924, −14.64615522257171952844736039514, −14.10862233589288455629614118862, −12.33954678408143323784139513844, −11.31538767511600109603705700156, −9.434420591874846042548239166354, −8.812643472837650067409725686787, −6.07889565597513368577601080630, −4.77002155527179426894864912161, −3.25960849837752797013653804291, 0.841618810941280350611269483588, 3.62450022703965031050779871180, 5.47693798597470759334405542154, 7.13221231607963175690306113095, 8.531632501272147968920210245166, 10.27039048652998048748959630616, 12.15086329126730888674533748452, 12.87709192204423397831630777483, 13.99609418752264290236294403721, 14.70835301571275326264406910870

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