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Properties

Label 1-185-185.43-r0-0-0
Degree $1$
Conductor $185$
Sign $-0.988 - 0.148i$
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  − 2-s − i·3-s + 4-s + i·6-s − i·7-s − 8-s − 9-s − 11-s − i·12-s − 13-s + i·14-s + 16-s + 17-s + 18-s − i·19-s + ⋯
L(s)  = 1  − 2-s − i·3-s + 4-s + i·6-s − i·7-s − 8-s − 9-s − 11-s − i·12-s − 13-s + i·14-s + 16-s + 17-s + 18-s − i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.988 - 0.148i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ -0.988 - 0.148i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03006053732 - 0.4020746814i\)
\(L(\frac12)\) \(\approx\) \(0.03006053732 - 0.4020746814i\)
\(L(1)\) \(\approx\) \(0.4523173885 - 0.2953422780i\)
\(L(1)\) \(\approx\) \(0.4523173885 - 0.2953422780i\)

Euler product

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

−27.49562843243845551818217207638, −26.919863079411097274218421944433, −25.858864183474014340816254223733, −25.227788139857352301262065514544, −24.08983837749383611183815666936, −22.75784091636716783390066989206, −21.520484045639041440835533435600, −21.03975717020393839554485641054, −19.94158306424968706296440130785, −18.90658491294328622640686948768, −18.0117289999391170398816049223, −16.886421371583221814091760985185, −16.048683499525179426805353709497, −15.259208215146544566680867257344, −14.38271744610663684229475702336, −12.36608659046001776130509723262, −11.62593900756482321996661380087, −10.20040057738924480676724638441, −9.84664733220104976438656921778, −8.55269082521536257705599442813, −7.7719308275271133226801309643, −6.02019792845060039387666450189, −5.1330241830899295592149493744, −3.27399981405854288427731638490, −2.20156467433446286568965060675, 0.40615098532015139404910910590, 1.911698344623384594031451355616, 3.12917963182490971554503288743, 5.28707043231882277900440029160, 6.71087564248010663389091488004, 7.51698800223818535031294486621, 8.20530226726737863186743987530, 9.69737960121018347455287613719, 10.63594214340415812180845634202, 11.74869390084325274117797775269, 12.75735118265312440585531226726, 13.88296334338980960809407199542, 15.05646919736376725911340270705, 16.45937730422145726361849445209, 17.19045671965996175477823325750, 18.088484248918115169695597052597, 18.90923321013270918876537906898, 19.915727792778322520197484289544, 20.45491331119768120990495151091, 21.88381051949255191016852579933, 23.505264379250535841518689303562, 23.89349099223423548349982900671, 24.961738809475192556676778466669, 26.00632917353156130837347750639, 26.50817975496199452088728661152

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