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Properties

Label 1-165-165.158-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: $\chi_{165} (158, \cdot )$
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

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

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