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Properties

Label 1-33-33.29-r0-0-0
Degree $1$
Conductor $33$
Sign $0.0694 + 0.997i$
Analytic cond. $0.153251$
Root an. cond. $0.153251$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s − 10-s + (−0.309 − 0.951i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.809 + 0.587i)19-s + (−0.309 − 0.951i)20-s − 23-s + (−0.809 − 0.587i)25-s + (0.809 − 0.587i)26-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s − 10-s + (−0.309 − 0.951i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.809 + 0.587i)19-s + (−0.309 − 0.951i)20-s − 23-s + (−0.809 − 0.587i)25-s + (0.809 − 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.0694 + 0.997i$
Analytic conductor: \(0.153251\)
Root analytic conductor: \(0.153251\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 33,\ (0:\ ),\ 0.0694 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5991850677 + 0.5589057585i\)
\(L(\frac12)\) \(\approx\) \(0.5991850677 + 0.5589057585i\)
\(L(1)\) \(\approx\) \(0.8640574851 + 0.5386211777i\)
\(L(1)\) \(\approx\) \(0.8640574851 + 0.5386211777i\)

Euler product

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

−36.44073407098404998105624896540, −35.12402860107431449486779685085, −33.42093103101159742881479384363, −32.04704484993487271052491475009, −31.2493670963927481351390892948, −30.108109802863686129576386912332, −28.47778691663878510319676603078, −28.02972055809608923101470948328, −26.585941912810171093502499733071, −24.4535104662030539498415734783, −23.717144714810940182209814426249, −21.97771759358303141429902456474, −21.00125171526141528992570144504, −19.874128287624809246252891428877, −18.58864920682525254661525156934, −17.107642095870645152471931089421, −15.247020648442842048439959754991, −13.7809728554709448627131936344, −12.29395921145039914952612623922, −11.43257097483626080621913372468, −9.53907377433332581456634288558, −8.27266993149060917433232802570, −5.45941421006020059718179406208, −4.14955408967141348675018529993, −1.83604896556715807157860586345, 3.46443914834755044140258535304, 5.25175213096146930586523251705, 7.06667982104237679791639364841, 8.03840493553050021811884341824, 10.16746384339313952135788027154, 11.90731959365218147885199693123, 13.81053467658681712632477585104, 14.6931364604418938453145561660, 15.98409846346874857105477559888, 17.5489904076273827198499840125, 18.49157705798670250377258503414, 20.436463572385459725974776351097, 22.11249348302195949606408878545, 23.033533718901334215289747995060, 24.21937309232051546558117073131, 25.502113354921091959189349860091, 26.81328962220430609735792584370, 27.45590308330601783673557214449, 29.80575435386051856892929916424, 30.698756266418211004451316712, 31.90013981208917764963585862660, 33.35142445465802441415520824264, 34.09634178940795593994869210105, 35.15455441508920977023051512993, 36.48476370303660356023432357091

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