↑

Properties

Label 1-185-185.12-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

Related objects

Downloads

Learn more

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: $\chi_{185} (12, \cdot )$
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 \( 1 + (0.866 - 0.5i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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