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Properties

Label 1-185-185.58-r1-0-0
Degree $1$
Conductor $185$
Sign $-0.946 - 0.321i$
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  + (−0.642 + 0.766i)2-s + (0.642 + 0.766i)3-s + (−0.173 − 0.984i)4-s − 6-s + (−0.342 + 0.939i)7-s + (0.866 + 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.642 − 0.766i)12-s + (0.984 − 0.173i)13-s + (−0.5 − 0.866i)14-s + (−0.939 + 0.342i)16-s + (0.984 + 0.173i)17-s + (−0.642 − 0.766i)18-s + (0.766 − 0.642i)19-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)2-s + (0.642 + 0.766i)3-s + (−0.173 − 0.984i)4-s − 6-s + (−0.342 + 0.939i)7-s + (0.866 + 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.642 − 0.766i)12-s + (0.984 − 0.173i)13-s + (−0.5 − 0.866i)14-s + (−0.939 + 0.342i)16-s + (0.984 + 0.173i)17-s + (−0.642 − 0.766i)18-s + (0.766 − 0.642i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.946 - 0.321i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ -0.946 - 0.321i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1905128379 + 1.153483488i\)
\(L(\frac12)\) \(\approx\) \(-0.1905128379 + 1.153483488i\)
\(L(1)\) \(\approx\) \(0.5874648250 + 0.6590603447i\)
\(L(1)\) \(\approx\) \(0.5874648250 + 0.6590603447i\)

Euler product

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

−26.29433973508016151384330393757, −25.913249137686531822512239424982, −24.711770630845374698305002173983, −23.58255682487293150835313141520, −22.66910879490630214185979830999, −21.14903966149687459340297262154, −20.543423112797552305147333868108, −19.66506368798756687167601196364, −18.68678032484842672571281821958, −18.2007632336074804786293844608, −16.82765402319056559009261720991, −16.03336519478683974277602697374, −14.13109679918429923390922092653, −13.523671962020221292801005690703, −12.572656970694787082060605575986, −11.44902955228259721492844001793, −10.342998601751204810766966517497, −9.29532754310581773173922543120, −8.10967336313494271515642535528, −7.49800457481788404436560420113, −6.08377082838130928298454955443, −3.841057311495326542509365975026, −3.11917657811925039071414100528, −1.572081677770704858677303228265, −0.47873700580775036471524048800, 1.83431253848641821564160620205, 3.35891456329010105194675841449, 4.983768470224575851141590081189, 5.840472749065682633335781940038, 7.4105978148608361518085661984, 8.37308005008172660609943710595, 9.34971735433587522662464068580, 10.031551425466676618121516238041, 11.237041696040858399944520725533, 12.907719142215157616058472369039, 14.12534089547056030447264515234, 15.08361532341110863590165724269, 15.77829136602236919374074487273, 16.46034105543018669360969913765, 17.93356058779986153247118537055, 18.63578881575730199706253908168, 19.77175418404588047443067584370, 20.587504935897381926614504888, 21.798531130423666901157875095422, 22.78003391539690338412337634740, 23.86434740538428084856241780937, 25.10804849443481780199943804383, 25.74569602038160419607870765720, 26.19386610351723962083336452981, 27.64619621179277914780964266742

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