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Properties

Label 1-165-165.59-r1-0-0
Degree $1$
Conductor $165$
Sign $0.999 + 0.0237i$
Analytic cond. $17.7317$
Root an. cond. $17.7317$
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 + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-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 + 23-s + (0.809 − 0.587i)26-s + (−0.309 + 0.951i)28-s + (0.809 − 0.587i)29-s + (0.309 + 0.951i)31-s + 32-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-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 + 23-s + (0.809 − 0.587i)26-s + (−0.309 + 0.951i)28-s + (0.809 − 0.587i)29-s + (0.309 + 0.951i)31-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.999 + 0.0237i$
Analytic conductor: \(17.7317\)
Root analytic conductor: \(17.7317\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 165,\ (1:\ ),\ 0.999 + 0.0237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.842202984 + 0.02189922591i\)
\(L(\frac12)\) \(\approx\) \(1.842202984 + 0.02189922591i\)
\(L(1)\) \(\approx\) \(1.180010477 + 0.3156524338i\)
\(L(1)\) \(\approx\) \(1.180010477 + 0.3156524338i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
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

−27.614893119725069111727354825, −26.83243495680684989844254730658, −25.52097622552621008047627282981, −24.218956406478270990165688160401, −23.55433176746190092545529767945, −22.37070488875462086967793106542, −21.32239578632439383017542722511, −20.97861580673976930855888011369, −19.5211187522528427653214144076, −18.857226261705920929791695381642, −17.821626195002359017496144288768, −16.749660841192857323870648030997, −15.00020496747456639811077963069, −14.50221901273906989929685545704, −13.20346542761596825317024240470, −12.17927629551741866778475626675, −11.35273895596605748412501472526, −10.31280878021595972422671289341, −9.090879794952387512823221569, −8.14898126453503682677819704259, −6.30168308430869655207464589874, −5.04969840672151703782358970207, −4.0388042415341970205823868748, −2.49820483734751186274472058341, −1.39666831725325105993889984514, 0.675309615403209746515152422862, 2.94401963387254687331720241075, 4.468633205926962461787359406790, 5.27064677035145147833242082612, 6.71979750019681919352456648956, 7.6815253889625976770405007652, 8.61560091843868512512955243955, 9.97019420836844925324660640830, 11.29640661899691209183358456495, 12.62652424648961540220737741404, 13.60820590606683415381143902321, 14.57262681070385785048833071420, 15.39408347671030352527395232758, 16.56954721245639941494306116515, 17.481287659055105882675767517307, 18.1719210670661173625338457347, 19.6094994413403636470873426338, 20.86219841276942533675034378079, 21.69826685353897400514180693792, 23.00751039202771952144827635642, 23.450834401524765493227148684554, 24.74657843366136013348333109870, 25.18340827252035669490682651391, 26.58269143641430988556633112200, 27.136583698082379899401535044

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