Properties

Label 2-33-3.2-c6-0-0
Degree $2$
Conductor $33$
Sign $0.676 - 0.736i$
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  − 11.1i·2-s + (−19.8 − 18.2i)3-s − 59.5·4-s + 223. i·5-s + (−202. + 221. i)6-s − 152.·7-s − 49.6i·8-s + (62.1 + 726. i)9-s + 2.48e3·10-s − 401. i·11-s + (1.18e3 + 1.08e3i)12-s − 1.70e3·13-s + 1.69e3i·14-s + (4.08e3 − 4.45e3i)15-s − 4.36e3·16-s + 8.37e3i·17-s + ⋯
L(s)  = 1  − 1.38i·2-s + (−0.736 − 0.676i)3-s − 0.930·4-s + 1.79i·5-s + (−0.939 + 1.02i)6-s − 0.443·7-s − 0.0970i·8-s + (0.0851 + 0.996i)9-s + 2.48·10-s − 0.301i·11-s + (0.685 + 0.629i)12-s − 0.777·13-s + 0.616i·14-s + (1.21 − 1.31i)15-s − 1.06·16-s + 1.70i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.676 - 0.736i$
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: \((2,\ 33,\ (\ :3),\ 0.676 - 0.736i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.426860 + 0.187571i\)
\(L(\frac12)\) \(\approx\) \(0.426860 + 0.187571i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (19.8 + 18.2i)T \)
11 \( 1 + 401. iT \)
good2 \( 1 + 11.1iT - 64T^{2} \)
5 \( 1 - 223. iT - 1.56e4T^{2} \)
7 \( 1 + 152.T + 1.17e5T^{2} \)
13 \( 1 + 1.70e3T + 4.82e6T^{2} \)
17 \( 1 - 8.37e3iT - 2.41e7T^{2} \)
19 \( 1 - 2.28e3T + 4.70e7T^{2} \)
23 \( 1 - 1.45e4iT - 1.48e8T^{2} \)
29 \( 1 + 2.21e4iT - 5.94e8T^{2} \)
31 \( 1 + 2.50e4T + 8.87e8T^{2} \)
37 \( 1 + 4.76e4T + 2.56e9T^{2} \)
41 \( 1 + 1.75e4iT - 4.75e9T^{2} \)
43 \( 1 - 369.T + 6.32e9T^{2} \)
47 \( 1 - 1.16e4iT - 1.07e10T^{2} \)
53 \( 1 - 1.91e5iT - 2.21e10T^{2} \)
59 \( 1 + 1.41e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.78e5T + 5.15e10T^{2} \)
67 \( 1 + 6.06e4T + 9.04e10T^{2} \)
71 \( 1 + 6.09e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.18e5T + 1.51e11T^{2} \)
79 \( 1 + 5.71e5T + 2.43e11T^{2} \)
83 \( 1 - 7.72e5iT - 3.26e11T^{2} \)
89 \( 1 + 8.08e5iT - 4.96e11T^{2} \)
97 \( 1 - 3.77e5T + 8.32e11T^{2} \)
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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

−15.45183437635426120104166637832, −13.97265698694692165987469809806, −12.79199491689037662202612159155, −11.64357240897938989135593388230, −10.79964133002693769899512341219, −9.976433126485766821866033641361, −7.38276142398257092837958896410, −6.16735120463469113674662556810, −3.45045013193053991542086731764, −2.00897666283200799471283384372, 0.25594883963179912511997457387, 4.71796603862152355342769553273, 5.37865647531059989663151712151, 7.03435935439247162208059399653, 8.739729224318508774075124123255, 9.673263187741776990449253237330, 11.77431353969897073613737687416, 12.86166900366599633244779928930, 14.45522868959483226940693219963, 15.89660519797823602565300971238

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