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Properties

Label 1-185-185.87-r0-0-0
Degree $1$
Conductor $185$
Sign $0.959 - 0.283i$
Analytic cond. $0.859136$
Root an. cond. $0.859136$
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  + (−0.939 − 0.342i)2-s + (−0.342 − 0.939i)3-s + (0.766 + 0.642i)4-s + i·6-s + (0.984 + 0.173i)7-s + (−0.5 − 0.866i)8-s + (−0.766 + 0.642i)9-s + (0.5 + 0.866i)11-s + (0.342 − 0.939i)12-s + (0.766 + 0.642i)13-s + (−0.866 − 0.5i)14-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + (0.939 − 0.342i)18-s + (0.342 + 0.939i)19-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.342 − 0.939i)3-s + (0.766 + 0.642i)4-s + i·6-s + (0.984 + 0.173i)7-s + (−0.5 − 0.866i)8-s + (−0.766 + 0.642i)9-s + (0.5 + 0.866i)11-s + (0.342 − 0.939i)12-s + (0.766 + 0.642i)13-s + (−0.866 − 0.5i)14-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + (0.939 − 0.342i)18-s + (0.342 + 0.939i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.959 - 0.283i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ 0.959 - 0.283i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7341742005 - 0.1061136530i\)
\(L(\frac12)\) \(\approx\) \(0.7341742005 - 0.1061136530i\)
\(L(1)\) \(\approx\) \(0.7079088913 - 0.1511617112i\)
\(L(1)\) \(\approx\) \(0.7079088913 - 0.1511617112i\)

Euler product

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

−27.0942569713600301735431169453, −26.719034444042492763850204724467, −25.54130323810650106403554675768, −24.4829683704038516164888159883, −23.66409536741845191259206785001, −22.441577460759229402644123004086, −21.32489880196805398381340007526, −20.45340336635528262076798042233, −19.67206916930591311432200307045, −18.06380515117943460982484296807, −17.69740198854951780352587876533, −16.45932179786169968832251510709, −15.83652355657665594749489560863, −14.79459231951228589878524693058, −13.86024075441835493197565998006, −11.75490351555692061401639493748, −11.0548745042781213548155466381, −10.308680195364141443545281429551, −8.91402492219535355503475209614, −8.403390033352109993823491176432, −6.82635311134441417151694069330, −5.68360033929736579209624103604, −4.58647778750504902023255348707, −2.91716736668036147019791984446, −0.960848802261530792490641096469, 1.44003285423419141366087450649, 2.11995458240231623536888494397, 4.06910944566146819185172417206, 5.88481323224742391195312564130, 6.97060679304233829922199528551, 7.96921318883155407597717136010, 8.79957310952759427179230912592, 10.20434400779253811317326218095, 11.476682986411664070630338441809, 11.85842335767032667779844272542, 13.08873943017253620833004023201, 14.340749681511559554103192832403, 15.650492958436297019913019256840, 16.95871417351690107007005732366, 17.64946367003780750204507177039, 18.344805491751151422539836858530, 19.28401622753642328464296379632, 20.232376698771394456548533283888, 21.17457383819596444772311714774, 22.34472099389303815412798039445, 23.615312193229608808539167003099, 24.47946626337686920111083400086, 25.28769798756379712866340435531, 26.17083029279790511829386895805, 27.462416942671377913102146377923

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