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Properties

Label 1-185-185.52-r0-0-0
Degree $1$
Conductor $185$
Sign $0.899 - 0.437i$
Analytic cond. $0.859136$
Root an. cond. $0.859136$
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  + (−0.766 − 0.642i)2-s + (0.642 + 0.766i)3-s + (0.173 + 0.984i)4-s − i·6-s + (0.342 − 0.939i)7-s + (0.5 − 0.866i)8-s + (−0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (−0.642 + 0.766i)12-s + (−0.173 − 0.984i)13-s + (−0.866 + 0.5i)14-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + (0.766 − 0.642i)18-s + (0.642 + 0.766i)19-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.642 + 0.766i)3-s + (0.173 + 0.984i)4-s − i·6-s + (0.342 − 0.939i)7-s + (0.5 − 0.866i)8-s + (−0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (−0.642 + 0.766i)12-s + (−0.173 − 0.984i)13-s + (−0.866 + 0.5i)14-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + (0.766 − 0.642i)18-s + (0.642 + 0.766i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.899 - 0.437i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ 0.899 - 0.437i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.009190992 - 0.2327158741i\)
\(L(\frac12)\) \(\approx\) \(1.009190992 - 0.2327158741i\)
\(L(1)\) \(\approx\) \(0.9511875000 - 0.1379616514i\)
\(L(1)\) \(\approx\) \(0.9511875000 - 0.1379616514i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (0.642 + 0.766i)T \)
7 \( 1 + (0.342 - 0.939i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.173 - 0.984i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (0.642 + 0.766i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + iT \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 - T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.342 + 0.939i)T \)
59 \( 1 + (-0.342 - 0.939i)T \)
61 \( 1 + (-0.984 + 0.173i)T \)
67 \( 1 + (-0.342 + 0.939i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.342 - 0.939i)T \)
83 \( 1 + (-0.984 - 0.173i)T \)
89 \( 1 + (0.342 + 0.939i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−27.06895589435116657665039087758, −26.12049094823259017309424831424, −25.40760747654367082043284464507, −24.54180768465872038513188581479, −23.98045220032302844519655914176, −22.81536336701759281260535889147, −21.37509697378750647443305163854, −20.17041177109696315410253511403, −19.33175010180948005349172887017, −18.54282379807014503664599819950, −17.74068759054266198617037600554, −16.78645411446527414115109771288, −15.27755800387833394504945133992, −14.80584301045380773358393540278, −13.76613242104494998494776226693, −12.36388200023622822228560023209, −11.425324621057373988116211231012, −9.74238218245469515348941674052, −8.95207606688127421093649932900, −8.08351211456533834401048178890, −6.96434289691820697156212199635, −6.13758243073874624471831903862, −4.59466836911135843608944743859, −2.48727408213290032097280105305, −1.49274826850382092551980259398, 1.20440530579781682804662361289, 2.99688392416747487210908974140, 3.70910989021222475673396303650, 5.130362112345306518366513761128, 7.21461917899420123250662731352, 8.11227919174078404866020008202, 9.10537557177132655940956419758, 10.159499050286744130126225864500, 10.83201501461709401067234671280, 11.95036189466682327186649529528, 13.48630965223452754084417417357, 14.1836685335411076328176514063, 15.67857363972117715396240763243, 16.53466122991678343364327356230, 17.4005611753790605904025887582, 18.59959712480627007587747600196, 19.75348716678115056356304622545, 20.24825757872031720797264468812, 21.14268427585445469725217207692, 22.017079217120998753449592046250, 23.07084918170088983442206389195, 24.79244505983723237569094505876, 25.37360311810173662269537282991, 26.67162154216486509655139193875, 27.11979771495714741384021089929

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