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Properties

Label 1-165-165.47-r0-0-0
Degree $1$
Conductor $165$
Sign $0.505 - 0.862i$
Analytic cond. $0.766256$
Root an. cond. $0.766256$
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  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (0.587 − 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.951 − 0.309i)13-s + (−0.809 + 0.587i)14-s + (0.309 + 0.951i)16-s + (0.951 − 0.309i)17-s + (0.809 − 0.587i)19-s + i·23-s + (0.809 + 0.587i)26-s + (0.951 − 0.309i)28-s + (−0.809 − 0.587i)29-s + (0.309 − 0.951i)31-s − i·32-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (0.587 − 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.951 − 0.309i)13-s + (−0.809 + 0.587i)14-s + (0.309 + 0.951i)16-s + (0.951 − 0.309i)17-s + (0.809 − 0.587i)19-s + i·23-s + (0.809 + 0.587i)26-s + (0.951 − 0.309i)28-s + (−0.809 − 0.587i)29-s + (0.309 − 0.951i)31-s − i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.505 - 0.862i$
Analytic conductor: \(0.766256\)
Root analytic conductor: \(0.766256\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 165,\ (0:\ ),\ 0.505 - 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6572950371 - 0.3767767666i\)
\(L(\frac12)\) \(\approx\) \(0.6572950371 - 0.3767767666i\)
\(L(1)\) \(\approx\) \(0.7225895055 - 0.2058252334i\)
\(L(1)\) \(\approx\) \(0.7225895055 - 0.2058252334i\)

Euler product

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

−27.74841901017362112956522880367, −26.9430613899722407034877112403, −26.01424964222805075442001474099, −24.86305377066328704889302174690, −24.433064133771687316442652916570, −23.23311269890300921111224743752, −21.84961648433582956662031236447, −20.88264431810018318094571890158, −19.85109164908332038764219531299, −18.75652676988895919374971739074, −18.14241811290481540682656623280, −16.98315848259080853420352206831, −16.17818727758433385384746053982, −14.899315800907230440149931120784, −14.364821080370077739530091430979, −12.39373776444181304780155433043, −11.60541121597508243444535841390, −10.341110442582031566515285959582, −9.36393499016922268846227767510, −8.2925192572168622746414384257, −7.37589070713610017419883787609, −6.019623497542598110262904697140, −4.962104747863321819774134992742, −2.83560788908913340036104885236, −1.51699886667592899425667240792, 0.954825526218223691676012424817, 2.49776721593212433418487496548, 3.92379793408610552860456432137, 5.551892902279384400209092786820, 7.36019003658813223628237280828, 7.68276757689795614435805482406, 9.26191480727244148995471646509, 10.099397172775631280134588421423, 11.19893917948893972233958860696, 12.06203487607093958299681928852, 13.3942988924412657052351694684, 14.667055486092438742023002028906, 15.83963644933002573120613309860, 17.00694878026926857371874644202, 17.55015046516044298911655030586, 18.689008996175857965431966246029, 19.71642909845033317124485831477, 20.493324248826158140140118356470, 21.38622750554204058728509620084, 22.56510849048898591215400489630, 23.928182868337021639129247760111, 24.72714745042153935903132744119, 25.84010182709027695773162629435, 26.74775159872391979899184431786, 27.43024157293634203398400599873

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