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Properties

Label 1-33-33.20-r1-0-0
Degree $1$
Conductor $33$
Sign $0.550 + 0.835i$
Analytic cond. $3.54634$
Root an. cond. $3.54634$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + 10-s + (−0.809 − 0.587i)13-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)17-s + (0.309 − 0.951i)19-s + (0.809 + 0.587i)20-s − 23-s + (0.309 − 0.951i)25-s + (−0.309 − 0.951i)26-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + 10-s + (−0.809 − 0.587i)13-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)17-s + (0.309 − 0.951i)19-s + (0.809 + 0.587i)20-s − 23-s + (0.309 − 0.951i)25-s + (−0.309 − 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.550 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.550 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.550 + 0.835i$
Analytic conductor: \(3.54634\)
Root analytic conductor: \(3.54634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 33,\ (1:\ ),\ 0.550 + 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.063390300 + 1.111528377i\)
\(L(\frac12)\) \(\approx\) \(2.063390300 + 1.111528377i\)
\(L(1)\) \(\approx\) \(1.656348695 + 0.6233009901i\)
\(L(1)\) \(\approx\) \(1.656348695 + 0.6233009901i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−36.373703686885210269012597534055, −34.15584855644664663730747387758, −33.35524141705024989689620072773, −32.27181578297308680816904871237, −30.82882768114773402517961095646, −29.78839209271969099469392033180, −29.06623520687862397455135332664, −27.41337654718705074162300331848, −25.95653559428545575165334581365, −24.42310016198412681689339018221, −23.240430315298461414951855194872, −21.9996022418500028826172702627, −20.99902105243617243485468823964, −19.69412217580757529953508872631, −18.2668376589502439870864140319, −16.65199603440251630554860597751, −14.57015811627787533903054182953, −13.97202428344093681946909259631, −12.406630877410720701597158241217, −10.77824142965780833880896719058, −9.8189813327586029784089497683, −7.12819269408254601491945241531, −5.5455797228575677479532303960, −3.74087944007447250429763122971, −1.81139567379962422737718613110, 2.539178522212275142217439983296, 4.90414958487731986711850017279, 5.93370034939438109247922932331, 7.87766181157032164057711400038, 9.45147661076103794659331847965, 11.79226584867006018092564270230, 12.93535965488956937937957050793, 14.27067493947605733986379607090, 15.55606479323997094867262045216, 16.94394580247123735357563444877, 18.10177599372759161311511017447, 20.31011774681807453359875511766, 21.50255261422482791984474742034, 22.3755671378883834686196293485, 24.12675736736728792064680575, 24.892057611768367569294092051683, 25.92560308935539658802204247965, 27.69642755919495394997266000062, 29.147583786469495413704779187593, 30.34208453954583497691365210756, 31.7833779883175742320955700467, 32.44611728451247118452870614901, 33.84323025904668261801945454965, 34.69420852105702217786390830321, 36.1468640483085427511637291648

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