Properties

Label 1-185-185.163-r0-0-0
Degree $1$
Conductor $185$
Sign $0.0103 + 0.999i$
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  + (0.766 + 0.642i)2-s + (−0.642 − 0.766i)3-s + (0.173 + 0.984i)4-s i·6-s + (−0.342 + 0.939i)7-s + (−0.5 + 0.866i)8-s + (−0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (0.642 − 0.766i)12-s + (0.173 + 0.984i)13-s + (−0.866 + 0.5i)14-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + (−0.766 + 0.642i)18-s + (0.642 + 0.766i)19-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (−0.642 − 0.766i)3-s + (0.173 + 0.984i)4-s i·6-s + (−0.342 + 0.939i)7-s + (−0.5 + 0.866i)8-s + (−0.173 + 0.984i)9-s + (0.5 − 0.866i)11-s + (0.642 − 0.766i)12-s + (0.173 + 0.984i)13-s + (−0.866 + 0.5i)14-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + (−0.766 + 0.642i)18-s + (0.642 + 0.766i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0103 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0103 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.0103 + 0.999i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ 0.0103 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8968177332 + 0.9061735284i\)
\(L(\frac12)\) \(\approx\) \(0.8968177332 + 0.9061735284i\)
\(L(1)\) \(\approx\) \(1.099624429 + 0.5112220127i\)
\(L(1)\) \(\approx\) \(1.099624429 + 0.5112220127i\)

Euler product

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

−27.34631401756622203636743962015, −26.177219832835390158836422716182, −24.89244370899192077233707936136, −23.66420284863541440875832999919, −22.80655310981307053340082962598, −22.44475491039242922937837200588, −21.28574494066075428505200478171, −20.20977630370112985555939551889, −19.90526234724114834384395133513, −18.15428196191684762334384928364, −17.27967143455232158334891686308, −15.95261739115853721063864188743, −15.28262951339039654288342626757, −14.06897268476230534144642903281, −13.084274659994519923057302160999, −11.91975039836300068282006318728, −11.11552519083440678389390059403, −10.02250102681113400085813769165, −9.492662806407260524134567855382, −7.27052999867788355400977673478, −6.12608362768173226485913473397, −4.930594381427675366563515059587, −4.08158382231758161585610912717, −2.95524873132855922139063865006, −0.8998265175637181612480867120, 1.93560552613477480697978842388, 3.43797658143237943996845583008, 4.949089860249297396000289009986, 6.149225305877684737744892950473, 6.51429514410763977746586250302, 8.01896863360708121073134450590, 8.91733170965552771315870191075, 10.88845448378333670225095229129, 12.03904907422330312767745584069, 12.4654007964860193067370983966, 13.74815453455075846391308196396, 14.47644642191362978283564295816, 15.99546739070922681764694583394, 16.492873159569965512100149375732, 17.64115144246627134490345996653, 18.63412733781783412232071329914, 19.54783354400820665413066222728, 21.225365744460881810928492388144, 21.99904092688421259276886824276, 22.68987097841737738344884114761, 23.83577717334997196545088166123, 24.40818016864218535104000820933, 25.222510289490799015870545529971, 26.19105890857961918751411812695, 27.42606143650736492273077092094

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