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Properties

Label 1-185-185.27-r1-0-0
Degree $1$
Conductor $185$
Sign $-0.965 + 0.261i$
Analytic cond. $19.8810$
Root an. cond. $19.8810$
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  + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s − 6-s + (−0.866 − 0.5i)7-s + i·8-s + (0.5 + 0.866i)9-s + 11-s + (0.866 − 0.5i)12-s + (−0.866 − 0.5i)13-s + 14-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.866 − 0.5i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s − 6-s + (−0.866 − 0.5i)7-s + i·8-s + (0.5 + 0.866i)9-s + 11-s + (0.866 − 0.5i)12-s + (−0.866 − 0.5i)13-s + 14-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.866 − 0.5i)18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.965 + 0.261i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ -0.965 + 0.261i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1055421709 + 0.7942165389i\)
\(L(\frac12)\) \(\approx\) \(0.1055421709 + 0.7942165389i\)
\(L(1)\) \(\approx\) \(0.6862001578 + 0.3570176320i\)
\(L(1)\) \(\approx\) \(0.6862001578 + 0.3570176320i\)

Euler product

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

−26.562181249033200813535089140635, −25.637830504987866099909154942002, −24.963627250833211287711187556029, −24.10661174117945542047684149644, −22.33881559721269842926481265326, −21.66124312281259077637148854620, −20.33774933386005197247890690510, −19.62363516990596646748178331283, −19.05423290291492257430505893737, −18.060981964900257332768990110659, −17.005265349212471889536820780836, −15.87189228430365328451791557951, −14.80642652688817486778971160518, −13.49462807839304450636901231199, −12.479715356949525544742265930869, −11.75079617782496850343226712058, −10.19846282423991573829253381066, −9.06935108458709368586871618893, −8.77523375878806613242640481211, −7.10346312742100014854740928784, −6.60722615458331471515245023218, −4.171486406701999057609006325323, −2.86642295474315959501295850537, −2.01288691531072362213084303220, −0.31905762092300023323050939668, 1.62426092982084887059057165604, 3.13205286681771158213419243696, 4.49041042535878055724140502172, 6.148142094025709436454573908295, 7.230750086386030147879761236200, 8.26053014005794715776539224720, 9.38024137398271954542425026237, 9.95923876215652231442670879677, 11.02776625132365901114914853692, 12.7005508918619882096046506351, 14.01160505916309665927784064832, 14.83648196709867729329681728113, 15.75451296570423104266683498374, 16.68606055288799660620357909832, 17.50948122523662654351225453454, 18.99121845068189255766945010236, 19.73464790712713492571321125876, 20.13859322850818635651934545912, 21.60991759652129911885898904560, 22.64363049464482512506241732520, 23.886669243699126065739834127465, 25.0862313764352918061938184477, 25.4387482154800953339690252446, 26.59213337564606739368299898810, 27.10642665965926684405119008442

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