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Properties

Label 2-33-11.10-c2-0-1
Degree $2$
Conductor $33$
Sign $0.297 - 0.954i$
Analytic cond. $0.899184$
Root an. cond. $0.948253$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.35i·2-s + 1.73·3-s − 1.53·4-s − 4.19·5-s + 4.07i·6-s − 6.42i·7-s + 5.79i·8-s + 2.99·9-s − 9.87i·10-s + (3.26 − 10.5i)11-s − 2.66·12-s − 7.68i·13-s + 15.1·14-s − 7.26·15-s − 19.7·16-s + 8.15i·17-s + ⋯
L(s)  = 1  + 1.17i·2-s + 0.577·3-s − 0.383·4-s − 0.839·5-s + 0.679i·6-s − 0.918i·7-s + 0.724i·8-s + 0.333·9-s − 0.987i·10-s + (0.297 − 0.954i)11-s − 0.221·12-s − 0.591i·13-s + 1.08·14-s − 0.484·15-s − 1.23·16-s + 0.479i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.297 - 0.954i$
Analytic conductor: \(0.899184\)
Root analytic conductor: \(0.948253\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1),\ 0.297 - 0.954i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.895316 + 0.659087i\)
\(L(\frac12)\) \(\approx\) \(0.895316 + 0.659087i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
11 \( 1 + (-3.26 + 10.5i)T \)
good2 \( 1 - 2.35iT - 4T^{2} \)
5 \( 1 + 4.19T + 25T^{2} \)
7 \( 1 + 6.42iT - 49T^{2} \)
13 \( 1 + 7.68iT - 169T^{2} \)
17 \( 1 - 8.15iT - 289T^{2} \)
19 \( 1 - 30.4iT - 361T^{2} \)
23 \( 1 + 31.6T + 529T^{2} \)
29 \( 1 + 6.88iT - 841T^{2} \)
31 \( 1 - 51.1T + 961T^{2} \)
37 \( 1 + 19.4T + 1.36e3T^{2} \)
41 \( 1 + 47.9iT - 1.68e3T^{2} \)
43 \( 1 - 81.8iT - 1.84e3T^{2} \)
47 \( 1 - 30.1T + 2.20e3T^{2} \)
53 \( 1 - 26.0T + 2.80e3T^{2} \)
59 \( 1 - 82.7T + 3.48e3T^{2} \)
61 \( 1 + 75.4iT - 3.72e3T^{2} \)
67 \( 1 + 34T + 4.48e3T^{2} \)
71 \( 1 + 72.7T + 5.04e3T^{2} \)
73 \( 1 + 54.8iT - 5.32e3T^{2} \)
79 \( 1 + 24.3iT - 6.24e3T^{2} \)
83 \( 1 + 0.923iT - 6.88e3T^{2} \)
89 \( 1 - 44.8T + 7.92e3T^{2} \)
97 \( 1 + 21.2T + 9.40e3T^{2} \)
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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

−16.42165756899425531894889209882, −15.68414411073736515058248920256, −14.50680690203436947147455460435, −13.69241088099027381857424138035, −11.89034134805745635211816149822, −10.35936293627991345156006638722, −8.292014088138413782435133884421, −7.71728914297544266680182657845, −6.12202236892873483528010606236, −3.89762662110820242516966203006, 2.44845411348976237297788982494, 4.24337549217304143668110691252, 7.05676282701268752472746213307, 8.813720025306646981678894787583, 9.988339563750427076946235572275, 11.67012015564178817791533468614, 12.13457659888978977829558713653, 13.59332694507980661338387411042, 15.22070072977571969719598611397, 15.87490538331399700014252366981

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