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Properties

Label 1-185-185.108-r1-0-0
Degree $1$
Conductor $185$
Sign $0.788 - 0.614i$
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  + (0.984 − 0.173i)2-s + (0.984 + 0.173i)3-s + (0.939 − 0.342i)4-s + 6-s + (0.642 − 0.766i)7-s + (0.866 − 0.5i)8-s + (0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (0.984 − 0.173i)12-s + (−0.342 − 0.939i)13-s + (0.5 − 0.866i)14-s + (0.766 − 0.642i)16-s + (−0.342 + 0.939i)17-s + (0.984 + 0.173i)18-s + (−0.173 + 0.984i)19-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (0.984 + 0.173i)3-s + (0.939 − 0.342i)4-s + 6-s + (0.642 − 0.766i)7-s + (0.866 − 0.5i)8-s + (0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (0.984 − 0.173i)12-s + (−0.342 − 0.939i)13-s + (0.5 − 0.866i)14-s + (0.766 − 0.642i)16-s + (−0.342 + 0.939i)17-s + (0.984 + 0.173i)18-s + (−0.173 + 0.984i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.788 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.788 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.788 - 0.614i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ 0.788 - 0.614i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.911573369 - 1.688046623i\)
\(L(\frac12)\) \(\approx\) \(4.911573369 - 1.688046623i\)
\(L(1)\) \(\approx\) \(2.739260223 - 0.5451924449i\)
\(L(1)\) \(\approx\) \(2.739260223 - 0.5451924449i\)

Euler product

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

−26.73743997661754922466908180973, −25.8549370877573628689711846397, −25.02618560118975030163635076614, −24.26986700142071722841773427601, −23.48849295796752494527346496743, −22.11524893741175182624810666482, −21.32540180788791720303624805384, −20.55566439959661182713257564696, −19.65003738716851312983347526106, −18.488277797038345029331352629323, −17.35306920278547499974490101613, −15.64453108094693586426392250813, −15.35525624314386280812951708657, −14.164166685057884691432879035574, −13.541897106869405462510692436, −12.29978504053305776870693516157, −11.57598396269767753468703543201, −9.94860731059132929789997736621, −8.69769042792634732869360964839, −7.583861721717456042567020307089, −6.69112748158670743814199814610, −5.07021878970153162345935236167, −4.22682247265738146791686265785, −2.61836686929861125049613869990, −1.9912679001856086318709816128, 1.36568589366880479775066131283, 2.74181088688541916514860094307, 3.79137815872265131267760959230, 4.78782305260443827033564562695, 6.169895765982302033422375841179, 7.65377927835099574049502091812, 8.3263762334051811388649906032, 10.26050755271622746257589288160, 10.674306248971216799663321732204, 12.26462388163864961347506077939, 13.28224687582475999202902744479, 14.05963601685384020406009617476, 14.836906499692926532209143769862, 15.78279898864471702058436405558, 16.8595390047497996813168059525, 18.417291765996715285171043526751, 19.63054815137926695036272301623, 20.25770308195523118820032536988, 21.14471915328246634269410258916, 21.85016123508805338482994652298, 23.11504272190774479292653324833, 24.12812591997999870924415460926, 24.70165869767361070583486423003, 25.83262524536172962522225582953, 26.75921983876042016650471713288

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