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Properties

Label 2-33-3.2-c2-0-4
Degree $2$
Conductor $33$
Sign $i$
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  − 3.31i·2-s + 3·3-s − 7·4-s + 6.63i·5-s − 9.94i·6-s − 8·7-s + 9.94i·8-s + 9·9-s + 22·10-s − 3.31i·11-s − 21·12-s + 4·13-s + 26.5i·14-s + 19.8i·15-s + 5.00·16-s − 13.2i·17-s + ⋯
L(s)  = 1  − 1.65i·2-s + 3-s − 1.75·4-s + 1.32i·5-s − 1.65i·6-s − 1.14·7-s + 1.24i·8-s + 9-s + 2.20·10-s − 0.301i·11-s − 1.75·12-s + 0.307·13-s + 1.89i·14-s + 1.32i·15-s + 0.312·16-s − 0.780i·17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $i$
Analytic conductor: \(0.899184\)
Root analytic conductor: \(0.948253\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1),\ i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.813379 - 0.813379i\)
\(L(\frac12)\) \(\approx\) \(0.813379 - 0.813379i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
11 \( 1 + 3.31iT \)
good2 \( 1 + 3.31iT - 4T^{2} \)
5 \( 1 - 6.63iT - 25T^{2} \)
7 \( 1 + 8T + 49T^{2} \)
13 \( 1 - 4T + 169T^{2} \)
17 \( 1 + 13.2iT - 289T^{2} \)
19 \( 1 + 6T + 361T^{2} \)
23 \( 1 + 6.63iT - 529T^{2} \)
29 \( 1 - 39.7iT - 841T^{2} \)
31 \( 1 + 26T + 961T^{2} \)
37 \( 1 - 30T + 1.36e3T^{2} \)
41 \( 1 + 13.2iT - 1.68e3T^{2} \)
43 \( 1 - 42T + 1.84e3T^{2} \)
47 \( 1 + 86.2iT - 2.20e3T^{2} \)
53 \( 1 - 59.6iT - 2.80e3T^{2} \)
59 \( 1 + 66.3iT - 3.48e3T^{2} \)
61 \( 1 - 12T + 3.72e3T^{2} \)
67 \( 1 - 2T + 4.48e3T^{2} \)
71 \( 1 - 59.6iT - 5.04e3T^{2} \)
73 \( 1 + 74T + 5.32e3T^{2} \)
79 \( 1 + 40T + 6.24e3T^{2} \)
83 \( 1 - 39.7iT - 6.88e3T^{2} \)
89 \( 1 - 119. iT - 7.92e3T^{2} \)
97 \( 1 - 62T + 9.40e3T^{2} \)
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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

−16.01398418524725385160950626319, −14.56671741647811408569175428269, −13.56469871033383997019074328678, −12.57513390229127967014706317425, −11.00398082884895023266604412421, −10.07999574161445947999091575522, −9.026354364107742844429786346354, −6.93970335167476522991480177841, −3.62327587915032566364728411320, −2.68129674928404865106074845948, 4.29685542700832488628962029342, 6.11198907827616590669215168131, 7.68385747545354478235407501398, 8.782121035391266122095343451094, 9.625344277890424690454082788157, 12.77656968448705242970921055972, 13.34101850161211493925304591171, 14.71618178752052947563904800675, 15.78213262934442172356563398244, 16.39340489348794926799824590827

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