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Properties

Label 2-33-3.2-c6-0-10
Degree $2$
Conductor $33$
Sign $0.892 - 0.450i$
Analytic cond. $7.59178$
Root an. cond. $2.75531$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.582i·2-s + (12.1 + 24.1i)3-s + 63.6·4-s − 147. i·5-s + (−14.0 + 7.08i)6-s + 326.·7-s + 74.3i·8-s + (−433. + 586. i)9-s + 86.1·10-s + 401. i·11-s + (773. + 1.53e3i)12-s + 2.27e3·13-s + 190. i·14-s + (3.56e3 − 1.79e3i)15-s + 4.03e3·16-s + 2.06e3i·17-s + ⋯
L(s)  = 1  + 0.0728i·2-s + (0.450 + 0.892i)3-s + 0.994·4-s − 1.18i·5-s + (−0.0650 + 0.0327i)6-s + 0.953·7-s + 0.145i·8-s + (−0.594 + 0.803i)9-s + 0.0861·10-s + 0.301i·11-s + (0.447 + 0.888i)12-s + 1.03·13-s + 0.0694i·14-s + (1.05 − 0.532i)15-s + 0.984·16-s + 0.420i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.892 - 0.450i$
Analytic conductor: \(7.59178\)
Root analytic conductor: \(2.75531\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :3),\ 0.892 - 0.450i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.39714 + 0.570043i\)
\(L(\frac12)\) \(\approx\) \(2.39714 + 0.570043i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-12.1 - 24.1i)T \)
11 \( 1 - 401. iT \)
good2 \( 1 - 0.582iT - 64T^{2} \)
5 \( 1 + 147. iT - 1.56e4T^{2} \)
7 \( 1 - 326.T + 1.17e5T^{2} \)
13 \( 1 - 2.27e3T + 4.82e6T^{2} \)
17 \( 1 - 2.06e3iT - 2.41e7T^{2} \)
19 \( 1 + 8.55e3T + 4.70e7T^{2} \)
23 \( 1 - 8.32e3iT - 1.48e8T^{2} \)
29 \( 1 + 4.05e4iT - 5.94e8T^{2} \)
31 \( 1 + 5.10e4T + 8.87e8T^{2} \)
37 \( 1 - 3.37e4T + 2.56e9T^{2} \)
41 \( 1 + 1.09e5iT - 4.75e9T^{2} \)
43 \( 1 + 3.92e4T + 6.32e9T^{2} \)
47 \( 1 - 3.53e4iT - 1.07e10T^{2} \)
53 \( 1 + 1.08e4iT - 2.21e10T^{2} \)
59 \( 1 - 1.55e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.03e5T + 5.15e10T^{2} \)
67 \( 1 + 3.92e5T + 9.04e10T^{2} \)
71 \( 1 - 2.60e5iT - 1.28e11T^{2} \)
73 \( 1 + 3.98e5T + 1.51e11T^{2} \)
79 \( 1 + 2.60e5T + 2.43e11T^{2} \)
83 \( 1 + 6.12e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.03e6iT - 4.96e11T^{2} \)
97 \( 1 - 8.25e5T + 8.32e11T^{2} \)
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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

−15.55118702100754988032182253647, −14.67104334843711285674820197767, −13.10743706343001763138794329543, −11.58763096099395745071283642981, −10.57401863231775442390719359396, −8.909291798122184186571375063780, −7.918059491632425199505165880838, −5.65536157026175076638177129008, −4.13310138613116925293833099922, −1.83203487601804793917005869549, 1.71172336929927488246455391699, 3.13018695803415051990316274356, 6.22836623798500363297632890962, 7.21221629155646334565193574704, 8.472988115252963920243718396255, 10.78208829361233628451628095810, 11.38894549641716644653178651660, 12.88606989152966398841498929487, 14.44262265009190486018565515155, 14.89085269666970075481134467250

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