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Properties

Label 1-185-185.62-r1-0-0
Degree $1$
Conductor $185$
Sign $0.834 + 0.551i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.834 + 0.551i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ 0.834 + 0.551i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.763253683 + 0.5299145594i\)
\(L(\frac12)\) \(\approx\) \(1.763253683 + 0.5299145594i\)
\(L(1)\) \(\approx\) \(1.117003022 + 0.08334596678i\)
\(L(1)\) \(\approx\) \(1.117003022 + 0.08334596678i\)

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

−26.82174643616286734828047139298, −26.15490566530011843514909514076, −25.215581318143252949261233671701, −24.1997101741583197448416419516, −23.5984435007854115863200830072, −21.61952592176126857301045662263, −20.82568904435948386781646034503, −20.13593180429921910385581002848, −19.03740976528123821050876948469, −18.426873051053811408900661670117, −17.11742709061217063925498710419, −16.20150850602153058420326117855, −15.29010050422819510570159013505, −14.15621142237931504990352101089, −13.41152055834383987979996836612, −11.51512550225067109302175378563, −10.74608117021372824663286349442, −9.51795764071917548505759068368, −8.7330951280169466703968652202, −7.68014724393854732060253183416, −6.93307465844026780575085388657, −5.141333936698750228855394370285, −3.54966746600540976205510868675, −2.23395536509801886987960925857, −0.86632131350282880841550302791, 1.40005070108806858430247074465, 2.382914843523705081412750951214, 3.60222968786701860600863722975, 5.50697759030170478442956851779, 7.09569568253750179850851965309, 8.07363974026866933677122587160, 8.69168391408279376292185843442, 9.876812177824169805165016692142, 10.77480617692980907042037394263, 12.28218252475525351577977786762, 12.92931601063156926008399502308, 14.80776620469947003368025817449, 15.09924408639840571207170414553, 16.36230568885123469817893445960, 17.76997155308183629569226503276, 18.38430749906445989653174227621, 19.217751091330210381263442996596, 20.451612790966311786407122316620, 20.7720419561331480316629091729, 21.905222264428064997900326628656, 23.56682669049496906485355233398, 24.68696554395988058549788908451, 25.34715409616307572716184237698, 25.98100119689466594571880363723, 27.162080361195566905842218156117

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